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Unformatted text preview: PHYSICS
PHYSICS B
PHYSICS C: MECHANICS
PHYSICS C: ELECTRICITY AND
MAGNETISM Course Description M AY 2 010, M AY 2 011 The College Board
The College Board is a notforproﬁt membership association whose mission is to connect students
to college success and opportunity. Founded in 1900, the association is composed of more than 5,600
schools, colleges, universities, and other educational organizations. Each year, the College Board
serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major
programs and services in college admissions, guidance, assessment, ﬁnancial aid, enrollment, and
teaching and learning. Among its bestknown programs are the SAT®, the PSAT/NMSQT®, and the
Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence
and equity, and that commitment is embodied in all of its programs, services, activities, and concerns.
For further information visit www.collegeboard.com.
The College Board and the Advanced Placement Program encourage teachers, AP Coordinators,
and school administrators to make equitable access a guiding principle for their AP programs. The
College Board is committed to the principle that all students deserve an opportunity to participate in
rigorous and academically challenging courses and programs. All students who are willing to accept
the challenge of a rigorous academic curriculum should be considered for admission to AP courses.
The Board encourages the elimination of barriers that restrict access to AP courses for students from
ethnic, racial, and socioeconomic groups that have been traditionally underrepresented in the AP
Program. Schools should make every effort to ensure that their AP classes reﬂect the diversity of their
student population. © 2009 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn
logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National
Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use
copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Contents
Welcome to the AP Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Course Audit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Exam Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Credit and Placement for AP Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Credit and Placement Policies for AP Grades . . . . . . . . . . . . . . . . . . . . . . 1
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3 AP Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
What We Are About: A Message from the Development Committee . . . . . . . . . 4
The Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Course Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Instructional Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Importance and Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Implementation and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Documenting Laboratory Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Physics B Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Physics C Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Comparison of Topics in Physics B and Physics C . . . . . . . . . . . . . . . . . . . . . . . 12
Content Outline for Physics B and Physics C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
The Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
The FreeResponse Sections—Student Presentation . . . . . . . . . . . . . . . . . . . . . . 18
Calculators and Equation Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Physics B Sample MultipleChoice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Answers to Physics B MultipleChoice Questions . . . . . . . . . . . . . . . . . . . . . . 29
Physics B Sample FreeResponse Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Physics C: Mechanics Sample MultipleChoice Questions . . . . . . . . . . . . . . . . . 35
Answers to Physics C: Mechanics MultipleChoice Questions . . . . . . . . . . . 39
Physics C: Mechanics Sample FreeResponse Questions . . . . . . . . . . . . . . . . . . 40
Physics C: Electricity and Magnetism Sample MultipleChoice
Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Answers to Physics C: Electricity and Magnetism
MultipleChoice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Physics C: Electricity and Magnetism Sample FreeResponse
Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Teacher Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Central (apcentral.collegeboard.com) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AP Publications and Other Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Teacher’s Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Course Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Released Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 54
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54 i Welcome to the AP® Program
For over 50 years, the College Board’s Advanced Placement Program (AP) has
partnered with colleges, universities, and high schools to provide students with the
opportunity to take collegelevel course work and exams while still in high school.
Offering more than 30 different subjects, each culminating in a rigorous exam, AP
provides motivated and academically prepared students with the opportunity to earn
college credit or placement and helps them stand out in the college admissions
process. Taught by dedicated, passionate AP teachers who bring cuttingedge content
knowledge and expert teaching skills to the classroom, AP courses help students
develop the study skills, habits of mind, and critical thinking skills that they will need
in college.
AP is accepted by more than 3,600 colleges and universities worldwide for college
credit, advanced placement, or both on the basis of successful AP Exam grades. This
includes over 90 percent of fouryear institutions in the United States.
More information about the AP Program is available at the back of this Course
Description and at AP Central®, the College Board’s online home for AP teachers
(apcentral.collegeboard.com). Students can ﬁnd more information at the AP student
site (www.collegeboard.com/apstudents). AP Courses
More than 30 AP courses in a wide variety of subject areas are now available. A
committee of college faculty and master AP teachers designs each AP course to cover
the information, skills, and assignments found in the corresponding college course. AP Exams
Each AP course has a corresponding exam that participating schools worldwide
administer in May. Except for AP Studio Art, which is a portfolio assessment, each AP
Exam contains a freeresponse section (essays, problem solving, oral responses, etc.)
as well as multiplechoice questions.
Written by a committee of college and university faculty and experienced AP
teachers, the AP Exam is the culmination of the AP course and provides students with
the opportunity to earn credit and/or placement in college. Exams are scored by
college professors and experienced AP teachers using scoring standards developed by
the committee. AP Course Audit
The intent of the AP Course Audit is to provide secondary and higher education
constituents with the assurance that an “AP” designation on a student’s transcript is
credible, meaning the AP Program has authorized a course that has met or exceeded
the curricular requirements and classroom resources that demonstrate the academic
rigor of a comparable college course. To receive authorization from the College Board
to label a course “AP,” teachers must participate in the AP Course Audit. Courses
authorized to use the “AP” designation are listed in the AP Course Ledger made
available to colleges and universities each fall. It is the school’s responsibility to ensure
that its AP Course Ledger entry accurately reﬂects the AP courses offered within each
academic year.
© 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 1 The AP Program unequivocally supports the principle that each individual school
must develop its own curriculum for courses labeled “AP.” Rather than mandating any
one curriculum for AP courses, the AP Course Audit instead provides each AP teacher
with a set of expectations that college and secondary school faculty nationwide have
established for collegelevel courses. AP teachers are encouraged to develop or maintain their own curriculum that either includes or exceeds each of these expectations;
such courses will be authorized to use the “AP” designation. Credit for the success of
AP courses belongs to the individual schools and teachers that create powerful, locally
designed AP curricula.
Complete information about the AP Course Audit is available at www.collegeboard
.com/apcourseaudit. AP Reading
AP Exams—with the exception of AP Studio Art, which is a portfolio assessment—
consist of dozens of multiplechoice questions scored by machine, and freeresponse
questions scored at the annual AP Reading by thousands of college faculty and expert
AP teachers. AP Readers use scoring standards developed by college and university
faculty who teach the corresponding college course. The AP Reading offers educators
both signiﬁcant professional development and the opportunity to network with
colleagues. For more information about the AP Reading, or to apply to serve as a
Reader, visit apcentral.collegeboard.com/readers. AP Exam Grades
The Readers’ scores on the freeresponse questions are combined with the results of
the computerscored multiplechoice questions; the weighted raw scores are summed
to give a composite score. The composite score is then converted to a grade on AP’s
5point scale:
AP GRADE
5
4
3
2
1 QUALIFICATION
Extremely well qualiﬁed
Well qualiﬁed
Qualiﬁed
Possibly qualiﬁed
No recommendation AP Exam grades of 5 are equivalent to A grades in the corresponding college course.
AP Exam grades of 4 are equivalent to grades of A–, B+, and B in college. AP Exam
grades of 3 are equivalent to grades of B–, C+, and C in college. 2 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Credit and Placement for AP Grades
Thousands of fouryear colleges grant credit, placement, or both for qualifying AP
Exam grades because these grades represent a level of achievement equivalent to that
of students who have taken the corresponding college course. This collegelevel
equivalency is ensured through several AP Program processes:
• College faculty are involved in course and exam development and other AP activities.
Currently, college faculty:
• Serve as chairs and members of the committees that develop the Course
Descriptions and exams in each AP course.
• Are responsible for standard setting and are involved in the evaluation of
student responses at the AP Reading. The Chief Reader for each AP subject is
a college faculty member.
• Lead professional development seminars for new and experienced AP teachers.
• Serve as the senior reviewers in the annual AP Course Audit, ensuring AP
teachers’ syllabi meet the curriculum guidelines of collegelevel courses.
• AP courses and exams are reviewed and updated regularly based on the results
of curriculum surveys at up to 200 colleges and universities, collaborations
among the College Board and key educational and disciplinary organizations, and
the interactions of committee members with professional organizations in their
discipline.
• Periodic college comparability studies are undertaken in which the performance of
college students on AP Exams is compared with that of AP students to conﬁrm that
the AP grade scale of 1 to 5 is properly aligned with current college standards.
For more information about the role of colleges and universities in the AP Program,
visit the Higher Ed Services section of the College Board Web site at professionals
.collegeboard.com/highered. Setting Credit and Placement Policies for AP Grades
The College Board Web site for education professionals has a section speciﬁcally for
colleges and universities that provides guidance in setting AP credit and placement
policies. Additional resources, including links to AP research studies, released exam
questions, and sample student responses at varying levels of achievement for each AP
Exam are also available. Visit professionals.collegeboard.com/highered/placement/ap.
The “AP Credit Policy Info” online search tool provides links to credit and placement policies at more than 1,000 colleges and universities. This tool helps students
ﬁnd the credit hours and/or advanced placement they may receive for qualifying exam
grades within each AP subject at a speciﬁed institution. AP Credit Policy Info is
available at www.collegeboard.com/ap/creditpolicy. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 3 AP Physics
INTRODUCTION What We Are About: A Message from the Development
Committee
The AP Physics Development Committee recognizes that curriculum, course content,
and assessment of scholastic achievement play complementary roles in shaping
education at all levels. The committee believes that assessment should support and
encourage the following broad instructional goals:
1. Physics knowledge—Basic knowledge of the discipline of physics, including
phenomenology, theories and techniques, concepts, and general principles
2. Problem solving—Ability to ask physical questions and to obtain solutions to
physical questions by use of qualitative and quantitative reasoning and by
experimental investigation
3. Student attributes—Fostering of important student attributes, including
appreciation of the physical world and the discipline of physics, curiosity,
creativity, and reasoned skepticism
4. Connections—Understanding connections of physics to other disciplines and to
societal issues
The ﬁrst three of these goals are appropriate for the AP and introductorylevel college
physics courses that should, in addition, provide a background for the attainment of
the fourth goal.
The AP Physics Exams have always emphasized achievement of the ﬁrst two goals.
Over the years, the deﬁnitions of basic knowledge of the discipline and problem
solving have evolved. The AP Physics courses have reﬂected changes in college
courses, consistent with our primary charge. We have increased our emphasis on
physical intuition, experimental investigation, and creativity. We include more openended questions in order to assess students’ ability to explain their understanding of
physical concepts. We structure questions that stress the use of mathematics to
illuminate the physical situation rather than to show manipulative abilities.
The committee is dedicated to developing exams that can be graded fairly and
consistently and that are free of ethnic, gender, economic, or other bias. We operate
under practical constraints of testing methods, allotted time, and large numbers of
students at widely spread geographical locations. In spite of these constraints, the
committee strives to design exams that promote excellent and appropriate instruction
in physics. THE COURSES
The AP Physics Exams are designed to test student achievement in the AP Physics
courses described in this book. These courses are intended to be representative of
courses commonly offered in colleges and universities, but they do not necessarily
correspond precisely to courses at any particular institution. The aim of an AP 4 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. secondary school course in physics should be to develop the students’ abilities to do
the following:
1. Read, understand, and interpret physical information—verbal, mathematical, and
graphical
2. Describe and explain the sequence of steps in the analysis of a particular
physical phenomenon or problem; that is,
a. describe the idealized model to be used in the analysis, including simplifying
assumptions where necessary;
b. state the concepts or deﬁnitions that are applicable;
c. specify relevant limitations on applications of these principles;
d. carry out and describe the steps of the analysis, verbally or mathematically;
and
e. interpret the results or conclusions, including discussion of particular cases
of special interest
3. Use basic mathematical reasoning—arithmetic, algebraic, geometric, trigonometric, or calculus, where appropriate—in a physical situation or problem
4. Perform experiments and interpret the results of observations, including making
an assessment of experimental uncertainties
In the achievement of these goals, concentration on basic principles of physics and
their applications through careful and selective treatment of wellchosen areas is more
important than superﬁcial and encyclopedic coverage of many detailed topics. Within
the general framework outlined on pages 13–15, teachers may exercise some freedom in
the choice of topics.
In the AP Physics Exams, an attempt is made through the use of multiplechoice
and freeresponse questions to determine how well these goals have been achieved by
the student either in a conventional course or through independent study. The level of
the student’s achievement is assigned an AP grade of 1 to 5, and many colleges use
this grade alone as the basis for placement and credit decisions.
Introductory college physics courses typically fall into one of three categories,
designated as A, B, and C in the following discussion.
Category A includes courses in which major concepts of physics are covered
without as much mathematical rigor as in more formal courses, such as Category B
and Category C, which are described below. The emphasis in Category A courses is on
developing a qualitative conceptual understanding of general principles and models
and on the nature of scientiﬁc inquiry. Some courses may also view physics primarily
from a cultural or historical perspective. Category A courses are generally intended for
students not majoring in a sciencerelated ﬁeld. The level of mathematical sophistication
usually includes some algebra and may extend to simple trigonometry, but rarely
beyond. These courses vary widely in content and approach, and at present there is
no AP course or exam in this category. A high school version of a Category A course
that concentrates on conceptual development and that provides an enriching laboratory
experience may be taken by students in the ninth or tenth grade and should provide
the ﬁrst course in physics that prepares them for a more mathematically rigorous AP
Physics B or C course. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 5 Category B courses build on the conceptual understanding attained in a ﬁrst course
in physics, such as the Category A course described above. These courses provide a
systematic development of the main principles of physics, emphasizing problem
solving and helping students develop a deep understanding of physics concepts. It is
assumed that students are familiar with algebra and trigonometry, although some
theoretical developments may use basic concepts of calculus. In most colleges, this is
a oneyear terminal course including a laboratory component and is not the usual
preparation for more advanced physics and engineering courses. However, Category B
courses often provide a foundation in physics for students in the life sciences, premedicine, and some applied sciences, as well as other ﬁelds not directly related to science.
AP Physics B is intended to be equivalent to such courses.
Category C courses also build on the conceptual understanding attained in a ﬁrst
course in physics, such as the Category A course described above. These courses
normally form the college sequence that serves as the foundation in physics for
students majoring in the physical sciences or engineering. The sequence is parallel to
or preceded by mathematics courses that include calculus. Methods of calculus are
used in formulating physical principles and in applying them to physical problems.
The sequence is more intensive and analytic than in Category B courses. Strong
emphasis is placed on solving a variety of challenging problems, some requiring
calculus, as well as continuing to develop a deep understanding of physics concepts. A
Category C sequence may be a very intensive oneyear course in college but often will
extend over one and onehalf to two years, and a laboratory component is also
included. AP Physics C is intended to be equivalent to part of a Category C sequence
and covers two major areas: mechanics, and electricity and magnetism, with equal
emphasis on both.
In certain colleges and universities, other types of unusually highlevel introductory
courses are taken by a few selected students. Selection of students for these courses is
often based on results of AP Exams, other college admission information, or a collegeadministered exam. The AP Exams are not designed to grant credit or exemption for
such highlevel courses but may facilitate admission to them. Course Selection
It is important for those teaching and advising AP students to consider the relation of
AP courses to a student’s college plans. In some circumstances it is advantageous to
take the AP Physics B course. The student may be interested in studying physics as a
basis for more advanced work in the life sciences, medicine, geology, and related
areas, or as a component in a nonscience college program that has science requirements. Credit or advanced placement for the Physics B course provides the student
with an opportunity either to have an accelerated college program or to meet a basic
science requirement; in either case the student’s college program may be enriched.
Access to an intensive physics sequence for physics or science majors is another
opportunity that may be available.
For students planning to specialize in a physical science or in engineering, most
colleges require an introductory physics sequence that includes courses equivalent to
Physics C. Since a previous or concurrent course in calculus is often required of
students taking Physics C, students who expect advanced placement or credit for 6 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. either Physics C exam should attempt an AP course in calculus as well; otherwise,
placement in the nextinsequence physics course may be delayed or even denied.
Either of the AP Calculus courses, Calculus AB or Calculus BC, should provide an
acceptable basis for students preparing to major in the physical sciences or engineering, but Calculus BC is recommended. Therefore, if such students must choose
between AP Physics or AP Calculus while in high school, they should probably choose
AP Calculus.
There are three separate AP Physics Exams, Physics B, Physics C: Mechanics, and
Physics C: Electricity and Magnetism. Each exam contains multiplechoice and freeresponse questions. The Physics B Exam is for students who have taken a Physics B
course or who have mastered the material of this course through independent study.
The Physics B Exam covers topics in mechanics, electricity and magnetism, ﬂuid
mechanics and thermal physics, waves and optics, and atomic and nuclear physics; a
single exam grade is reported. Similarly, the two Physics C Exams correspond to the
Physics C course sequence. One exam covers mechanics; the other covers electricity
and magnetism. Students may take either or both exams, and separate grades are
reported.
Further descriptions of the AP Physics courses and their corresponding exams in
terms of topics, level, mathematical rigor, and typical textbooks are presented in the
pages that follow. Information about organizing and conducting AP Physics courses,
of interest to both beginning and experienced AP teachers, may be found in the
AP Physics Teacher’s Guide. This publication includes practical advice from successful
AP teachers. The 2004 AP Physics B and Physics C Released Exams book contains the
complete exams, with solutions and grading standards for the freeresponse sections
and sample student responses, as well as statistical data on student performance. For
information about ordering these publications and others, see page 54. Additional useful
information may be found at AP Central (apcentral.collegeboard.com). Instructional Approaches
It is strongly recommended that both Physics B and Physics C be taught as
secondyear physics courses. A ﬁrstyear physics course aimed at developing a
thorough understanding of important physical principles and that permits students to
explore concepts in the laboratory provides a richer experience in the process of
science and better prepares them for the more analytical approaches taken in AP
courses.
However, secondary school programs for the achievement of AP course goals can
take other forms as well, and the imaginative teacher can design approaches that best
ﬁt the needs of his or her students. In some schools, AP Physics has been taught
successfully as a very intensive ﬁrstyear course; but in this case there may not be
enough time to cover the material in sufﬁcient depth to reinforce the students’
conceptual understanding or to provide adequate laboratory experiences. This
approach can work for highly motivated, able students but is not generally recommended. Independent study or other ﬁrstyear physics courses supplemented with
extra work for individual motivated students are also possibilities that have been
successfully implemented. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 7 If AP Physics is taught as a secondyear course, it is recommended that the course
meet for at least 250 minutes per week (the equivalent of a 50minute period every
day). However, if it is to be taught as a ﬁrstyear course, approximately 90 minutes per
day (450 minutes per week) is recommended in order to devote sufﬁcient time to
study the material to an appropriate depth and allow time for labs.
In a school that uses block scheduling, it is strongly recommended that AP Physics B
be scheduled to extend over an entire year. A oneyear AP course should not be taught
in one semester, as this length of time is insufﬁcient for students to properly assimilate
and understand the important concepts of physics that are covered in the syllabus.
Each of the Physics C courses, but not both, can be taught in one semester.
More detailed descriptions about alternate approaches can be found in the Teacher’s
Guide. Whichever approach is taken, the nature of the AP course requires teachers to
spend time on the extra preparation needed for both class and laboratory. AP teachers
should have a teaching load that is adjusted accordingly. Laboratory
Importance and Rationale
Laborator y experience must be part of the education of AP Physics students
and should be included in all AP Physics courses, just as it is in introductory
college physics courses. In textbooks and problems, most attention is paid to
idealized situations: friction is often assumed to be constant or absent; meters read
true values; heat insulators are perfect; gases follow the ideal gas equation. It is in the
laboratory that the validity of these assumptions can be questioned, because there the
student meets nature as it is rather than in idealized form. Consequently, AP students
should be able to:
• design experiments;
• observe and measure real phenomena;
• organize, display, and critically analyze data;
• analyze sources of error and determine uncertainties in measurement;
• draw inferences from observations and data; and
• communicate results, including suggested ways to improve experiments and
proposed questions for further study.
Laboratory experience is also important in helping students understand the topics
being considered. Thus it is valuable to ask students to write informally about what
they have done, observed, and concluded, as well as for them to keep wellorganized
laboratory notebooks.
Students need to be proﬁcient in problem solving and in the application of
fundamental principles to a wide variety of situations. Problemsolving ability can be
fostered by investigations that are somewhat nonspeciﬁc. Such investigations are often
more interesting and valuable than “cookbook” experiments that merely investigate a
wellestablished relationship and can take important time away from the rest of the
course. 8 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Some questions or parts of questions on each AP Physics Exam deal with labrelated skills, such as design of experiments, data analysis, and error analysis, and
may distinguish between students who have had laboratory experience and those who
have not. In addition, understanding gained in the laboratory may improve students’
test performance overall. Implementation and Recommendations
Laboratory programs in both college courses and AP courses differ widely, and there
is no clear evidence that any one approach is necessarily best. This diversity of
approaches should be encouraging to the high school teacher of an AP course. The
success of a given program depends strongly on the interests and enthusiasm of the
teacher and on the general ability and motivation of the students involved.
Although programs differ, the AP Physics Development Committee has made some
recommendations in regard to school resources and scheduling. Since an AP
course is a college course, the equipment and time allotted to laboratories
should be similar to that in a college course. Therefore, school administrators
should realize the implications, in both cost and time, of incorporating
serious laboratories into their program. Schools must ensure that students
have access to scientiﬁc equipment and all materials necessar y to conduct
handson, collegelevel physics laborator y investigations as outlined in the
teacher’s course syllabus.
In addition to equipment commonly included in college labs, students in AP Physics
should have adequate and timely access to computers that are connected to the Internet
and its many online resources. Students should also have access to computers with
appropriate sensing devices and software for use in gathering, graphing, and analyzing
laboratory data and writing reports. Although using computers in this way is a useful
activity and is encouraged, some initial experience with gathering, graphing, and
manipulating data by hand is also important so that students attain a better feel for the
physical realities involved in the experiments. And it should be emphasized that simulating an experiment on a computer cannot adequately replace the actual, handson
experience of doing an experiment.
Flexible or modular scheduling is best in order to meet the time requirements
identiﬁed in the course outline. Some schools are able to assign daily double periods
so that laboratory and quantitative problemsolving skills may be fully developed.
A weekly extended or double laboratory period is recommended for labs. It is not
advisable to attempt to complete highquality AP laboratory work entirely within
standard 45 to 50minute periods.
If AP Physics is taught as a secondyear physics course, the AP labs should build on
and extend the lab experiences of the ﬁrstyear course. The important criterion is that
students completing an AP Physics course must have had laboratory experiences that
are roughly equivalent to those in a comparable introductory college course.
Past surveys of introductory college physics courses, both noncalculus and
calculusbased, have revealed that on average about 20 percent of the total course
credit awarded can be attributed to lab performance; from two to three hours per
week are typically devoted to laboratory activities. Secondary schools may have © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 9 difﬁculty scheduling this much weekly time for lab. However, the college academic
year typically contains fewer weeks than the secondary school year, so AP teachers
may be able to schedule a few more lab periods during the year than can colleges.
Also, college faculty have reported that some lab time occasionally may be used for
other purposes. Nevertheless, in order for AP students to have sufﬁcient time for lab,
at least one double or extended period per week is recommended for all AP Physics
courses.
Laboratory activities in colleges and AP courses can involve different levels of
student involvement. They can generally be classiﬁed as: (1) prescribed or “cookbook,”
(2) limited investigations with some direction provided, and (3) open investigations
with little or no direction provided. While many college professors believe that labs in
the latter two categories have more value to students, they report often being limited
in their ability to institute them by large class sizes and other factors. In this respect,
AP teachers often have an advantage in being able to offer more openended labs to
their students.
In past surveys, colleges have cited use of the following techniques to assess
student lab performance: lab reports, direct observation, written tests designed
speciﬁcally for lab, labrelated questions on regular lecture tests, lab practical exams,
and maintenance of lab notebooks. When the colleges assessed laboratory skills with
written test questions, they reported attempting to assess the following skills in order
of decreasing frequency: analysis of data, analysis of errors, design of experiments,
and evaluation of experiments and suggestions for future investigations.
A more detailed laboratory guide is available and can be ordered through AP Central.
This guide contains descriptions of a number of experiments that typify the type and
level of skills that should be developed by AP students in conducting laboratory
investigations. The experiments are not mandatory; they can be modiﬁed or similar
experiments substituted as long as they assist the student in developing these skills.
The AP Physics Teacher’s Guide also provides additional suggestions for the laboratory.
The guide mentions speciﬁc experiments that other AP teachers have tried and liked
and lists publications and other sources of information that may provide additional
ideas for lowcost experiments. It will be helpful to experienced AP teachers as well as
to those just beginning to teach courses in AP Physics. Documenting Laboratory Experience
The laboratory is important for both AP and college students. Students who have had
laboratory experience in high school will be in a better position to validate their AP
courses as equivalent to the corresponding college courses and to undertake the
laboratory work in more advanced courses with greater conﬁdence. Most college
placement policies assume that students have had laboratory experience, and students
should be prepared to show evidence of their laboratory work in case the college asks
for it. Such experience should be documented for the AP course by keeping a lab
notebook or a portfolio of lab reports. Students should be encouraged to keep copies
of this work and any other work from previous lab experience. Presenting evidence of
adequate collegelevel laboratory experience to the colleges they attend, as an adjunct
to their AP grades, can be very useful to students if they desire credit for or exemption
from an introductory college course that includes a laboratory. Although colleges can
expect that most entering AP students have been exposed to many of the same 10 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. laboratory experiments performed by their own introductory students, individual
consultation with students is often used to help determine the nature of their
laboratory experience. Physics B Course
The Physics B course includes topics in both classical and modern physics. A
knowledge of algebra and basic trigonometry is required for the course; the basic
ideas of calculus may be introduced in connection with physical concepts, such as
acceleration and work. Understanding of the basic principles involved and the ability
to apply these principles in the solution of problems should be the major goals of the
course. Consequently, the course should utilize guided inquiry and studentcentered
learning to foster the development of critical thinking skills.
Physics B should provide instruction in each of the following ﬁve content areas:
Newtonian mechanics, ﬂuid mechanics and thermal physics, electricity and magnetism,
waves and optics, and atomic and nuclear physics. A content outline and percentage
goals for covering each major topic in the exam are on pages 13–15. A more detailed
topic outline is contained in the “Learning Objectives for AP Physics,” which can be
found on AP Central.
Many colleges and universities include additional topics in their survey courses.
Some AP teachers may wish to add supplementary material to a Physics B course.
Many teachers have found that a good time to do this is late in the year, after the
AP Exams have been given.
The Physics B course should also include a handson laboratory component
comparable to introductory collegelevel physics laboratories, with a minimum of 12
studentconducted laboratory investigations representing a variety of topics covered in
the course. Each student should complete a lab notebook or portfolio of lab reports.
The school should ensure that each student has a copy of a collegelevel textbook
(supplemented when necessary to meet the curricular requirements) for individual
use inside and outside of the classroom. A link to a list of examples of acceptable
textbooks can be found on the Physics B course home page on the AP Central Web
site. The AP Physics Teacher’s Guide includes some additional suggestions for
supplementary books and other materials. Physics C Courses
There are two AP Physics C courses—Physics C: Mechanics and Physics C: Electricity
and Magnetism, each corresponding to approximately a semester of college work.
Mechanics is typically taught ﬁrst, and some AP teachers may choose to teach this
course only. If both courses are taught over the course of a year, approximately equal
time should be given to each. Both courses should utilize guided inquiry and studentcentered learning to foster the development of critical thinking skills and should use
introductory differential and integral calculus throughout the course.
Physics C: Mechanics should provide instruction in each of the following six
content areas: kinematics; Newton’s laws of motion; work, energy, and power; systems
of particles and linear momentum; circular motion and rotation; and oscillations and
gravitation.
© 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 11 Physics C: Electricity and Magnetism should provide instruction in each of the
following ﬁve content areas: electrostatics; conductors, capacitors, and dielectrics;
electric circuits; magnetic ﬁelds; and electromagnetism.
Content outlines for both courses and percentage goals for covering each major
topic in the exams are on pages 13–15. A more detailed topic outline is contained in
the “Learning Objectives for AP Physics,” which can be found on AP Central.
Most colleges and universities include in similar courses additional topics such as
wave motion, kinetic theory and thermodynamics, optics, alternating current circuits,
or special relativity. Although wave motion, optics, and kinetic theory and thermodynamics are usually the most commonly included, there is little uniformity among
such offerings, and these topics are not included in the Physics C Exams. The
Development Committee recommends that supplementary material be added to
Physics C when it is possible to do so. Many teachers have found that a good time to
do this is late in the year, after the AP Exams have been given.
Each Physics C course should also include a handson laboratory component
comparable to a semesterlong introductory collegelevel physics laboratory. Students
should spend a minimum of 20 percent of instructional time engaged in handson
laboratory work. Each student should complete a lab notebook or portfolio of lab
reports.
The school should ensure that each student has a calculusbased collegelevel
textbook (supplemented when necessary to meet the curricular requirements) for
individual use inside and outside of the classroom. A link to lists of examples of
acceptable textbooks can be found on the Physics C course home pages on the
AP Central Web site. The AP Physics Teacher’s Guide includes some additional
suggestions for supplementary books and other materials. Comparison of Topics in Physics B and Physics C
To serve as an aid for devising AP Physics courses and to more clearly identify the
speciﬁcs of the exams, a detailed topical structure has been developed that relies
heavily on information obtained in college surveys. The general areas of physics are
subdivided into major categories on pages 13–15, and for each category the percentage
goals for each exam are given. These goals should serve only as a guide and should
not be construed as reﬂecting the proportion of course time that should be devoted to
each category.
Also, for each major category, some important subtopics are listed. The checkmarks
indicate the subtopics that may be covered in each exam. Questions for the exam will
come from these subtopics, but not all of the subtopics will necessarily be included in
every exam, just as they are not necessarily included in every AP or college course.
It should be noted that although fewer topics are covered in Physics C than in
Physics B, they are covered in greater depth and with greater analytical and mathematical sophistication, including calculus applications. 12 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Content Outline for Physics B and Physics C
A more detailed topic outline is contained in the “Learning Objectives for
AP Physics,” which can be found on AP Central. Content Area Percentage Goals for Exams
Physics B
Physics C:
Mechanics I. Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Kinematics (including vectors, vector algebra,
components of vectors, coordinate systems,
displacement, velocity, and acceleration)
1. Motion in one dimension
2. Motion in two dimensions, including
projectile motion 35% 100% 7% 18% √
√ √
√ B. Newton’s laws of motion
1. Static equilibrium (ﬁrst law)
2. Dynamics of a single particle (second law)
3. Systems of two or more objects (third law) 9%
√
√
√ 20%
√
√
√ C. Work, energy, power
1. Work and work–energy theorem
2. Forces and potential energy
3. Conservation of energy
4. Power 5%
√
√
√
√ 14%
√
√
√
√ D. Systems of particles, linear momentum
1. Center of mass
2. Impulse and momentum
3. Conservation of linear momentum,
collisions 4% 12%
√
√
√ √
√ E. Circular motion and rotation
1. Uniform circular motion
2. Torque and rotational statics
3. Rotational kinematics and dynamics
4. Angular momentum and its conservation 4%
√
√ 18%
√
√
√
√ F. Oscillations and gravitation
1. Simple harmonic motion (dynamics and
energy relationships)
2. Mass on a spring
3. Pendulum and other oscillations
4. Newton’s law of gravity
5. Orbits of planets and satellites
a. Circular
b. General 6%
√ 18%
√ √
√
√ √
√
√ √ √
√ © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 13 Percentage Goals for Exams
Physics B Content Area II. Fluid Mechanics and Thermal Physics. . . . . . . . . . . . . . . . . 15% A. Fluid Mechanics
1. Hydrostatic pressure
2. Buoyancy
3. Fluid ﬂow continuity
4. Bernoulli’s equation 6%
√
√
√
√ B. Temperature and heat
1. Mechanical equivalent of heat
2. Heat transfer and thermal expansion 2%
√
√ C. Kinetic theory and thermodynamics
1. Ideal gases
a. Kinetic model
b. Ideal gas law
2. Laws of thermodynamics
a. First law (including processes on
pV diagrams)
b. Second law (including heat engines) 7%
√
√
√
√
Physics C:
Electricity and
Magnetism III. Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . 25% 100% A. Electrostatics
1. Charge and Coulomb’s law
2. Electric ﬁeld and electric potential (including
point charges)
3. Gauss’s law
4. Fields and potentials of other charge distributions 5%
√
√ 30%
√
√ B. Conductors, capacitors, dielectrics
1. Electrostatics with conductors
2. Capacitors
a. Capacitance
b. Parallel plate
c. Spherical and cylindrical
3. Dielectrics 4%
√ 14%
√ √
√ √
√
√
√ 7%
√
√ 20%
√
√ √ √
√ C. Electric circuits
1. Current, resistance, power
2. Steadystate direct current circuits with
batteries and resistors only
3. Capacitors in circuits
a. Steady state
b. Transients in RC circuits 14 √
√ © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Content Area Percentage Goals for Exams
Physics B
Physics C:
Electricity and
Magnetism D. Magnetic Fields
1. Forces on moving charges in magnetic ﬁelds
2. Forces on currentcarrying wires in
magnetic ﬁelds
3. Fields of long currentcarrying wires
4. Biot–Savart law and Ampere’s law 4%
√
√ 20%
√
√ √ √
√ E. Electromagnetism
1. Electromagnetic induction (including
Faraday’s law and Lenz’s law)
2. Inductance (including LR and LC circuits)
3. Maxwell’s equations 5%
√ 16%
√ IV. Waves and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √
√
15% A. Wave motion (including sound)
1. Traveling waves
2. Wave propagation
3. Standing waves
4. Superposition 5%
√
√
√
√ B. Physical optics
1. Interference and diffraction
2. Dispersion of light and the electromagnetic
spectrum 5%
√
√ C. Geometric optics
1. Reﬂection and refraction
2. Mirrors
3. Lenses 5%
√
√
√ V. Atomic and Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . .
A. Atomic physics and quantum effects
1. Photons, the photoelectric effect,
Compton scattering, xrays
2. Atomic energy levels
3. Waveparticle duality
B. Nuclear physics
1. Nuclear reactions (including conservation
of mass number and charge)
2. Mass–energy equivalence 10%
7%
√
√
√
3%
√
√ © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 15 Laboratory and experimental situations: Each exam will include one or more questions
or parts of questions posed in a laboratory or experimental setting. These questions are
classiﬁed according to the content area that provides the setting for the situation, and
each content area may include such questions. These questions generally assess some
understanding of content as well as experimental skills, as described on the following
pages.
Miscellaneous: Each exam may include occasional questions that overlap several
major topical areas or questions on miscellaneous topics such as identiﬁcation of
vectors and scalars, vector mathematics, graphs of functions, history of physics, or
contemporary topics in physics. THE EXAMS
The AP Physics B Exam is 3 hours long, divided equally between a 70question
multiplechoice section and a freeresponse section. The two sections are weighted
equally, and a single grade is reported for the B Exam.
The freeresponse section will usually contain 6 or 7 questions. Examples of
possible formats are 2 questions of about 17 minutes each and 5 shorter questions
of about 11 minutes each, or 4 questions of about 17 minutes each and 2 shorter
questions of about 11 minutes each. However, future exams might include a
combination of questions of other lengths.
Each Physics C Exam is 1 hour and 30 minutes long. A student may take either or
both exams, and separate grades are reported for each. The time for each exam is
divided equally between a 35question multiplechoice section and a freeresponse
section; the two sections are weighted equally in the determination of each grade. The
usual format for each freeresponse section has been 3 questions, each taking about
15 minutes. However, future exams might include a larger number of shorter
questions.
The percentages of each exam devoted to each major category are speciﬁed in the
preceding pages. Departures from these percentages in the freeresponse section in
any given year are compensated for in the multiplechoice section so that the overall
topic distribution for the entire exam is achieved as closely as possible, although it
may not be reached exactly.
Some questions, particularly in the freeresponse sections, may involve topics from
two or more major categories. For example, a question may utilize a setting involving
principles from electricity and magnetism or atomic and nuclear physics, but parts of
the question may also involve the application of principles of mechanics to this setting,
either alone or in combination with the principles from electricity and magnetism or
atomic and nuclear physics. Such a question would not be classiﬁed uniquely according to any particular topic but would receive partial classiﬁcations by topics in
proportion to the principles needed to arrive at the answers.
On both exams the multiplechoice section emphasizes the breadth of the students’
knowledge and understanding of the basic principles of physics; the freeresponse
section emphasizes the application of these principles in greater depth in solving more
extended problems. In general, questions may ask students to:
• determine directions of vectors or paths of particles;
• draw or interpret diagrams; 16 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. • interpret or express physical relationships in graphical form;
• account for observed phenomena;
• interpret experimental data, including their limitations and uncertainties;
• construct and use conceptual models and explain their limitations;
• explain steps taken to arrive at a result or to predict future physical behavior;
• manipulate equations that describe physical relationships;
• obtain reasonable estimates; or
• solve problems that require the determination of physical quantities in either
numerical or symbolic form and that may require the application of single or
multiple physical concepts.
Laboratoryrelated questions may ask students to:
• design experiments, including identifying equipment needed and describing
how it is to be used, drawing diagrams or providing descriptions of experimental
setups, or describing procedures to be used, including controls and measurements to be taken;
• analyze data, including displaying data in graphical or tabular form, ﬁtting lines
and curves to data points in graphs, performing calculations with data, or making
extrapolations and interpolations from data;
• analyze errors, including identifying sources of errors and how they propagate,
estimating magnitude and direction of errors, determining signiﬁcant digits, or
identifying ways to reduce errors; or
• communicate results, including drawing inferences and conclusions from
experimental data, suggesting ways to improve experiments, or proposing
questions for further study.
The freeresponse section of each exam is printed in a separate booklet in which each
part of a question is followed by a blank space for the student’s solution. The same
questions without the blank answer spaces are printed on green paper as an insert in
the exam booklet. This green insert also contains a Table of Information and tables of
commonly used equations. The Table of Information, which is also printed near the
front of each multiplechoice section, includes numerical values of some physical
constants and conversion factors and states some conventions used in the exams. The
equation tables are described in greater detail in a later section. The green insert can
be removed from the freeresponse answer booklet and used for reference when
answering the freeresponse questions only.
The International System of Units (SI) is used predominantly in both exams. The
use of rulers or straightedges is permitted on the freeresponse sections to facilitate
the sketching of graphs or diagrams that might be required in these sections.
Since the complete exams are intended to provide the maximum information about
differences in students’ achievement in physics, students may ﬁnd them more difﬁcult © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 17 than many classroom exams. The best way for teachers to familiarize their students
with the level of difﬁculty is to give them actual released exams (both multiplechoice
and freeresponse sections) from past administrations. Information about ordering
publications is on page 54. Recent freeresponse sections can also be found on AP
Central, along with scoring guidelines and some sample student responses. The FreeResponse Sections—Student Presentation
Students are expected to show their work in the spaces provided for the solution for
each part of a freeresponse question. If they need more space, they should clearly
indicate where the work is continued or they may lose credit for it. If students make a
mistake, they may cross it out or erase it. Crossedout work and any work shown on
the green insert will not be scored, and credit may be lost for incorrect work that is
not crossed out.
In scoring the freeresponse sections, credit for the answers depends on the quality
of the solutions and the explanations given; partial solutions may receive partial credit,
so students are advised to show all their work. Correct answers without supporting
work may lose credit. This is especially true when students are asked speciﬁcally to
justify their answers, in which case the Exam Readers are looking for some verbal or
mathematical analysis that shows how the students arrived at their answers. Also, all
ﬁnal numerical answers should include appropriate units.
On the AP Physics Exams the words “justify,” “explain,” “calculate,” “what is,”
“determine,” “derive,” “sketch,” and “plot” have precise meanings. Students should pay
careful attention to these words in order to obtain maximum credit and should avoid
including irrelevant or extraneous material in their answers.
The ability to justify an answer in words shows understanding of the principles
underlying physical phenomena in addition to the ability to perform the mathematical
manipulations necessary to generate a correct answer. Students will be directed to
justify or explain their answers on many of the questions they encounter on the AP
Physics Exams. The words “justify” and “explain” indicate that the student should
support the answer with prose, equations, calculations, diagrams, or graphs. The prose
or equations may in some cases refer to fundamental ideas or relations in physics,
such as Newton’s laws, conservation of energy, Gauss’s law, or Bernoulli’s equation. In
other cases, the justiﬁcation or explanation may take the form of analyzing the behavior
of an equation for large or small values of a variable in the equation.
The words “calculate,” “what is,” “determine,” and “derive” have distinct meanings
on the AP Physics Exams. “Calculate” means that a student is expected to show work
leading to a ﬁnal answer, which may be algebraic but more often is numerical. “What
is” and “determine” indicate that work need not necessarily be explicitly shown to
obtain full credit. Showing work leading to answers is a good idea, as it may earn a
student partial credit in the case of an incorrect answer, but this step may be skipped
by the conﬁdent or harried student. “Derive” is more speciﬁc and indicates that the
students need to begin their solutions with one or more fundamental equations, such
as those given on the AP Physics Exam equation sheet. The ﬁnal answer, usually
algebraic, is then obtained through the appropriate use of mathematics. 18 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. The words “sketch” and “plot” relate to studentproduced graphs. “Sketch” means
to draw a graph that illustrates key trends in a particular relationship, such as slope,
curvature, intercept(s), or asymptote(s). Numerical scaling or speciﬁc data points are
not required in a sketch. “Plot” means to draw the data points given in the problem on
the grid provided, either using the given scale or indicating the scale and units when
none are provided.
Additional information about study skills and testtaking strategies can be found at
AP Central. Calculators and Equation Tables
Policies regarding the use of calculators on the exams take into account the expansion
of the capabilities of scientiﬁc calculators, which now include not only programming
and graphing functions but also the availability of stored equations and other data. For
taking the sections of the exams in which calculators are permitted, students should
be allowed to use the calculators to which they are accustomed, except as noted
below.* On the other hand, they should not have access to information in their
calculators that is not available to other students, if that information is needed to
answer the questions.
Calculators are NOT permitted on the multiplechoice sections of the
Physics B and Physics C exams. The purpose of the multiplechoice sections is to
assess the breadth of students’ knowledge and understanding of the basic concepts
of physics. The multiplechoice questions emphasize conceptual understanding and
qualitative applications. However, many physical deﬁnitions and principles are quantitative by nature and can therefore be expressed as equations. The knowledge of these
basic deﬁnitions and principles, expressed as equations, is a part of the content of
physics that should be learned by physics students and will continue to be assessed in
the multiplechoice sections. However, any numeric calculations using these equations
required in the multiplechoice sections will be kept simple. Also, in some questions,
the answer choices differ by several orders of magnitude so that the questions can
be answered by estimation. Students should be encouraged to develop their skills
not only in estimating answers but also in recognizing answers that are physically
unreasonable or unlikely.
Calculators are allowed on the freeresponse section of all exams. Any
programmable or graphing calculator may be used except as noted below,*
and students will not be required to erase their calculator memories before
and after the exam. The freeresponse sections emphasize solving indepth problems
where knowledge of which principles to apply and how to apply them is the most
important aspect of the solution to these problems.
Regardless of the type of calculator allowed, the exams are designed and scored to
minimize the necessity of doing lengthy computations. When freeresponse problems * Exceptions to calculator use. Calculators that are not permitted are PowerBooks and portable/handheld
computers; electronic writing pads or peninput/stylusdriven devices (e.g., Palm, PDAs, Casio ClassPad 300);
pocket organizers; models with QWERTY (i.e., typewriter) keypads (e.g., TI92 Plus, Voyage 200); models with
paper tapes; models that make noise or “talk”; models that require an electrical outlet; cell phone calculators.
Students may not share calculators. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 19 involve calculations, most of the points awarded in the grading of the solution are
given for setting up the solution correctly rather than for actually carrying out the
computation.
Tables containing commonly used physics equations are printed in the
green insert provided with each exam for students to use when taking the
freeresponse section. The equation tables may NOT be used when taking the
multiplechoice section. The Table of Information and the equation tables for the 2010
and 2011 exams are included as an insert in this book so that they can easily be
removed and duplicated for use by students. In general, the tables for each year’s
exam will be printed and distributed with the Course Description at least a year in
advance so that students can become accustomed to using them throughout the year.
However, since the equations will be provided with the exams, students are NOT
allowed to bring their own copies to the exam room.
One of the purposes of providing the commonly used equations is to make the freeresponse sections equitable for those students who do not have access to equations
stored in their calculators. The availability of these equations means that in the scoring
of the freeresponse sections little or no credit will be awarded for simply writing down
correct equations or for ambiguous answers unsupported by explanations or logical
development.
The equations in the tables express relationships that are encountered most
frequently in AP Physics courses and exams. However, they do not include all
equations that might possibly be used. For example, they do not include many
equations that can be derived by combining others in the tables. Nor do they include
equations that are simply special cases of any that are in the tables. Students are
responsible for understanding the physical principles that underlie each equation
and for knowing the conditions for which each equation is applicable.
The equations are grouped in tables according to major content category. Within
each table, the symbols used for the variables in that table are deﬁned. However, in
some cases the same symbol is used to represent different quantities in different
tables. It should be noted that there is no uniform convention among textbooks for
the symbols used in writing equations. The equation tables follow many common
conventions, but in some cases consistency was sacriﬁced for the sake of clarity.
In summary, the purpose of minimizing numerical calculations in both sections of
the exams and providing equations with the freeresponse sections is to place greater
emphasis on the understanding and application of fundamental physical principles and
concepts. For solving problems, a sophisticated programmable or graphing calculator,
or the availability of stored equations, is no substitute for a thorough grasp of the
physics involved. 20 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B Physics B Sample MultipleChoice Questions
Most of the following sample questions, illustrative of the Physics B Exam, have
appeared in past exams. The answers are on page 29. Additional questions can be
found in the 2004 AP Physics B and Physics C Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations listed in the table of information included with the exams (see insert in this book.)
To simplify calculations, you may use g = 10 m/s2 in all problems.
Directions: Each of the questions or incomplete statements below is followed by ﬁve
suggested answers or completions. Select the one that is best in each case. 1. An object is thrown with a horizontal velocity of 20 m/s from a cliff that is 125 m
above level ground. If air resistance is negligible, the time that it takes the object to
fall to the ground from the cliff is most nearly
(A) 3 s
(B) 5 s
(C) 6 s
(D) 12 s
(E) 25 s 2. The motion of a particle along a straight line is represented by the position versus
time graph above. At which of the labeled points on the graph is the magnitude of
the acceleration of the particle greatest?
(A)
(B)
(C)
(D)
(E) A
B
C
D
E © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 21 Sample Questions for Physics B Questions 3–4 A 2 kg block, starting from rest, slides 20 m down a frictionless inclined plane from X to
Y, dropping a vertical distance of 10 m as shown above.
3. The magnitude of the net force on the block while it is sliding is most nearly
(A)
(B)
(C)
(D)
(E) 4. The speed of the block at point Y is most nearly
(A)
(B)
(C)
(D)
(E) 22 10.1 N
10.4 N
12.5 N
15.0 N
10.0 N 107 m/s
110 m/s
114 m/s
120 m/s
100 m/s © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B 5. A block of mass 2 kg slides along a horizontal tabletop. A horizontal applied force
of 12 N and a vertical applied force of 15 N act on the block, as shown above. If
the coefficient of kinetic friction between the block and the table is 0.2, the
frictional force exerted on the block is most nearly
(A)
(B)
(C)
(D)
(E) 6. 1N
3N
4N
5N
7N A ball of mass M and speed v collides headon with a ball of mass 2M and speed
v , as shown above. If the two balls stick together, their speed after the collision is
2
(A) 0
(B) v
2
(C) ͌2v (D) ͌3v 2 2
(E) 3v
2 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 23 Sample Questions for Physics B 7. A massless rigid rod of length 3d is pivoted at a fixed point W, and two forces each
of magnitude F are applied vertically upward as shown above. A third vertical force
of magnitude F may be applied, either upward or downward, at one of the labeled
points. With the proper choice of direction at each point, the rod can be in
equilibrium if the third force of magnitude F is applied at point
(A)
(B)
(C)
(D)
(E) 8. W only
Y only
V or X only
V or Y only
V W, or X
, An ideal monatomic gas is compressed while its temperature is held constant. What
happens to the internal energy of the gas during this process, and why?
(A) It decreases because the gas does work on its surroundings.
(B) It decreases because the molecules of an ideal gas collide.
(C) It does not change because the internal energy of an ideal gas depends only on
its temperature.
(D) It increases because work is done on the gas.
(E) It increases because the molecules travel a shorter path between collisions. 24 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B 9. In the pV diagram above, the initial state of a gas is shown at point X. Which of the
curves represents a process in which no work is done on or by the gas?
(A)
(B)
(C)
(D)
(E) XA
XB
XC
XD
XE P• •q T•
10. An isolated positive charge q is in the plane of the page, as shown above. The
directions of the electric field vectors at points P and T, which are also in the plane
of the page, are given by which of the following?
(A)
(B)
(C)
(D)
(E) Point P
Left
Right
Left
Right
Left Point T
Right
Left
Toward the top of the page
Toward the top of the page
Toward the bottom of the page © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 25 Sample Questions for Physics B Questions 11–12 relate to the following circuit in which the battery has zero internal
resistance. 11. What is the current in the 4 Ω resistor while the switch S is open?
(A)
(B)
(C)
(D)
(E) 0A
0.6 A
1.2 A
2.0 A
3.0 A 12. When the switch S is closed and the 10 µF capacitor is fully charged, what is the
voltage across the capacitor?
(A)
(B)
(C)
(D)
(E) 110 V
116 V
112 V
160 V
120 V Flow 1 • • 2 13. A fluid flows steadily from left to right in the pipe shown above. The diameter of
the pipe is less at point 2 than at point 1, and the fluid density is constant throughout
the pipe. How do the velocity of flow and the pressure at points 1 and 2 compare?
(A)
(B)
(C)
(D)
(E) 26 Velocity
v1 < v2
v1 < v2
v1 = v2
v1 > v2
v1 > v2 Pressure
p1 = p2
p1 > p2
p1 < p2
p1 = p2
p1 > p2 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B 14. Two long parallel wires, separated by a distance d, carry equal currents I toward the
top of the page, as shown above. The magnetic field due to the wires at a point
halfway between them is
(A)
(B)
(C)
(D)
(E) zero in magnitude
directed into the page
directed out of the page
directed to the right
directed to the left 15. A source S of sound and a listener L each can be at rest or can move directly toward
or away from each other with speed v0. In which of the following situations will the
observer hear the lowest frequency of sound from the source?
(A) S
•
v= 0 (B) S
•
v=0 (C) S • vϭv0
(D) S • vϭv0
(E) L
•
v= 0
L
•ৎ
vϭv0
L
•
vϭ0 L
•ৎ
vϭv0 S
L
•ৎ •
vϭv0 vϭv0 16. The wavelength of yellow sodium light in vacuum is 5.89 ϫ 10–7 m. The speed of
this light in glass with an index of refraction of 1.5 is most nearly
(A)
(B)
(C)
(D)
(E) 4 ϫ 10–7 m/s
9 ϫ 10–7 m/s
2 ϫ 108 m/s
3 ϫ 108 m/s
4 ϫ 108 m/s © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 27 Sample Questions for Physics B 17. An object O is in front of a convex mirror. The focal point of the mirror is labeled
F and the center of curvature is labeled C. The direction of the reflected ray is
correctly illustrated in all of the following EXCEPT which diagram? 18. A system initially consists of an electron and an incident photon. The electron and
the photon collide, and afterward the system consists of the electron and a scattered
photon. The electron gains kinetic energy as a result of this collision. Compared
with the incident photon, the scattered photon has
(A)
(B)
(C)
(D)
(E) 28 the same energy
a smaller speed
a larger speed
a smaller frequency
a larger frequency © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B 19. In an experiment, light of a particular wavelength is incident on a metal surface,
and electrons are emitted from the surface as a result. To produce more electrons
per unit time but with less kinetic energy per electron, the experimenter should do
which of the following?
(A)
(B)
(C)
(D)
(E) Increase the intensity and decrease the wavelength of the light.
Increase the intensity and the wavelength of the light.
Decrease the intensity and the wavelength of the light.
Decrease the intensity and increase the wavelength of the light.
None of the above would produce the desired result. 20. When 10B is bombarded by neutrons, a neutron can be absorbed and an alpha
particle (4He) emitted. The kinetic energy of the reaction products is equal to the
(A)
(B)
(C)
(D) kinetic energy of the incident neutron
total energy of the incident neutron
energy equivalent of the mass decrease in the reaction
energy equivalent of the mass decrease in the reaction, minus the kinetic
energy of the incident neutron
(E) energy equivalent of the mass decrease in the reaction, plus the kinetic energy
of the incident neutron Answers to Physics B MultipleChoice Questions
1–B 5–E 9–B 13 – B 17 – D 2–C 6–A 10 – E 14 – A 18 – D 3–E 7–C 11 – B 15 – D 19 – B 4–C 8–C 12 – B 16 – C 20 – E © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 29 Sample Questions for Physics B Physics B Sample FreeResponse Questions
The following six questions constituted the complete freeresponse section of the 2006
AP Physics B Exam. All freeresponse questions released since 1999 can be found at
AP Central.
Directions: Answer all six questions, which are weighted according to the points
indicated. The suggested times are about 17 minutes for answering each of Questions 1–4
and about 11 minutes for answering each of Questions 5–6. The parts within a question
may not have equal weight. Show all your work in the pink booklet in the spaces
provided after each part, NOT in this green insert. 1. (15 points)
An ideal spring of unstretched length 0.20 m is placed horizontally on a frictionless
table as shown above. One end of the spring is fixed and the other end is attached
to a block of mass M = 8.0 kg. The 8.0 kg block is also attached to a massless
string that passes over a small frictionless pulley. A block of mass m = 4.0 kg hangs
from the other end of the string. When this springandblocks system is in
equilibrium, the length of the spring is 0.25 m and the 4.0 kg block is 0.70 m above
the floor.
(a) On the figures below, draw freebody diagrams showing and labeling the
forces on each block when the system is in equilibrium.
M = 8.0 kg m = 4.0 kg (b) Calculate the tension in the string.
(c) Calculate the force constant of the spring.
The string is now cut at point P.
(d) Calculate the time taken by the 4.0 kg block to hit the floor.
(e) Calculate the frequency of oscillation of the 8.0 kg block.
(f) Calculate the maximum speed attained by the 8.0 kg block.
30 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B 2. (15 points)
A worldclass runner can complete a 100 m dash in about 10 s. Past studies have
shown that runners in such a race accelerate uniformly for a time tu and then run at
constant speed for the remainder of the race. A worldclass runner is visiting your
physics class. You are to develop a procedure that will allow you to determine the
uniform acceleration au and an approximate value of tu for the runner in a 100 m
dash. By necessity your experiment will be done on a straight track and include
your whole class of eleven students.
(a) By checking the line next to each appropriate item in the list below, select the
equipment, other than the runner and the track, that your class will need to do
the experiment.
____ Stopwatches ____ Tape measures ____ Rulers ____ Masking tape ____ Metersticks ____ Starter’s pistol ____ String ____ Chalk (b) Outline the procedure that you would use to determine au and tu, including a
labeled diagram of the experimental setup. Use symbols to identify carefully
what measurements you would make and include in your procedure how you
would use each piece of the equipment you checked in part (a).
(c) Outline the process of data analysis, including how you will identify the
portion of the race that has uniform acceleration, and how you would calculate
the uniform acceleration. 3. (15 points)
Two point charges, q1 and q2, are placed 0.30 m apart on the xaxis, as shown in the
figure above. Charge q1 has a value of −3.0 ϫ 10Ϫ9 C . The net electric field at
point P is zero.
(a) What is the sign of charge q2?
____Positive ____ Negative Justify your answer.
(b) Calculate the magnitude of charge q2.
(c) Calculate the magnitude of the electric force on q2 and indicate its direction.
(d) Determine the xcoordinate of the point on the line between the two charges at
which the electric potential is zero.
(e) How much work must be done by an external force to bring an electron from
infinity to the point at which the electric potential is zero? Explain your
reasoning. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 31 Sample Questions for Physics B 4. (15 points)
A student performs an experiment to determine the index of refraction n of a
rectangular glass slab in air. She is asked to use a laser beam to measure angles of
incidence θ i in air and corresponding angles of refraction θ r in glass. The
measurements of the angles for five trials are given in the table below.
Trial θi θr 1 30º 20º 2 40º 27º 3 50º 32º 4 60º 37º 5 70º 40º (a) Complete the last two columns in the table by calculating the quantities that
need to be graphed to provide a linear relationship from which the index of
refraction can be determined. Label the top of each column.
(b) On the grid below, plot the quantities calculated in (a) and draw an appropriate
graph from which the index of refraction can be determined. Label the axes. 32 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics B (c) Using the graph, calculate the index of refraction of the glass slab. The student is also asked to determine the thickness of a film of oil (n = 1.43) on
the surface of water (n = 1.33). Light from a variable wavelength source is incident
vertically onto the oil film as shown above. The student measures a maximum
in the intensity of the reflected light when the incident light has a wavelength of
600 nm.
(d) At which of the two interfaces does the light undergo a 180º phase change on
reflection?
___The airoil interface only ___The oilwater interface only ___Both interfaces ___Neither interface (e) Calculate the minimum possible thickness of the oil film. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 33 Sample Questions for Physics B 5. (10 points)
A cylinder with a movable frictionless piston contains an ideal gas that is initially in
state 1 at 1 ϫ 105 Pa, 373 K, and 0.25 m3. The gas is taken through a reversible
thermodynamic cycle as shown in the PV diagram above.
(a) Calculate the temperature of the gas when it is in the following states.
i. State 2
ii. State 3
(b) Calculate the net work done on the gas during the cycle.
(c) Was heat added to or removed from the gas during the cycle?
____ Added ____ Removed ____ Neither added nor removed Justify your answer.
6. (10 points)
A photon with a wavelength of 1.5 ϫ 10Ϫ8 m is emitted from an ultraviolet source
into a vacuum.
(a) Calculate the energy of the photon.
(b) Calculate the de Broglie wavelength of an electron with kinetic energy equal to
the energy of the photon.
(c) Describe an experiment that illustrates the wave properties of this electron. 34 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Mechanics Physics C: Mechanics Sample MultipleChoice Questions
Most of the following sample questions have appeared in past exams. The answers are on
page 39. Additional questions can be found in the 2004 AP Physics B and Physics C
Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations
listed in the table of information included with the exams (see insert in this book). To
simplify calculations, you may use g = 10 m/s2 in all problems.
Directions: Each of the questions or incomplete statements below is followed by ﬁve
suggested answers or completions. Select the one that is best in each case.
Questions 1–2
The speed v of an automobile moving on a straight road is given in meters per second as
a function of time t in seconds by the following equation:
v = 4 + 2t3 1. What is the acceleration of the automobile at t = 2 s?
(A)
(B)
(C)
(D)
(E) 2. 12 m/s2
16 m/s2
20 m/s2
24 m/s2
28 m/s2 How far has the automobile traveled in the interval between t = 0 and t = 2 s?
(A)
(B)
(C)
(D)
(E) 16 m
20 m
24 m
32 m
72 m © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 35 Sample Questions for Physics C: Mechanics 3. If a particle moves in a plane so that its position is described by the functions
x = A cos t and y = A sin t, the particle is
(A)
(B)
(C)
(D)
(E) 4. moving with constant speed along a circle
moving with varying speed along a circle
moving with constant acceleration along a straight line
moving along a parabola
oscillating back and forth along a straight line A system in equilibrium consists of an object of weight W that hangs from three
ropes, as shown above. The tensions in the ropes are T1, T2, and T3. Which of the
following are correct values of T2 and T3?
T2 T3 (A) W tan 60° W
cos 60° (B) W tan 60° W
sin 60° (C) W tan 60° W sin 60° (D) W
cos 60° (E) 36 W
tan 60°
W
tan 60° W
sin 60° © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Mechanics 5. The constant force F with components Fx = 3 N and Fy = 4 N, shown above, acts
on a body while that body moves from the point P (x = 2 m, y = 6 m) to the point
Q (x = 14 m, y = 1 m). How much work does the force do on the body during this
process?
(A)
(B)
(C)
(D)
(E) 6. 16 J
30 J
46 J
56 J
65 J The sum of all the external forces on a system of particles is zero. Which of the
following must be true of the system?
(A)
(B)
(C)
(D)
(E) The total mechanical energy is constant.
The total potential energy is constant.
The total kinetic energy is constant.
The total linear momentum is constant.
It is in static equilibrium. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 37 Sample Questions for Physics C: Mechanics 7. A toy cannon is fixed to a small cart and both move to the right with speed v along
a straight track, as shown above. The cannon points in the direction of motion.
When the cannon fires a projectile the cart and cannon are brought to rest. If M is
the mass of the cart and cannon combined without the projectile, and m is the mass
of the projectile, what is the speed of the projectile relative to the ground
immediately after it is fired?
(A) Mv
m (B) (M + m)v
m
(C) (M – m)v
m
(D) mv
M
(E) 8. A disk X rotates freely with angular velocity on frictionless bearings, as shown
above. A second identical disk Y, initially not rotating, is placed on X so that both
disks rotate together without slipping. When the disks are rotating together, which
of the following is half what it was before?
(A)
(B)
(C)
(D)
(E) 38 mv
(M – m) Moment of inertia of X
Moment of inertia of Y
Angular velocity of X
Angular velocity of Y
Angular momentum of both disks © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Mechanics 9. The ring and the disk shown above have identical masses, radii, and velocities, and
are not attached to each other. If the ring and the disk each roll without slipping up
an inclined plane, how will the distances that they move up the plane before coming
to rest compare?
(A)
(B)
(C)
(D)
(E) The ring will move farther than will the disk.
The disk will move farther than will the ring.
The ring and the disk will move equal distances.
The relative distances depend on the angle of elevation of the plane.
The relative distances depend on the length of the plane. 10. Let g be the acceleration due to gravity at the surface of a planet of radius R. Which
of the following is a dimensionally correct formula for the minimum kinetic energy
K that a projectile of mass m must have at the planet’s surface if the projectile is to
escape from the planet’s gravitational field?
(A) K = √gR
(B) K = mgR
(C) K = mg
R
(D) K = m g
R (E) K = gR Answers to Physics C: Mechanics MultipleChoice Questions
1–D 3–A 5–A 7–B 9–A 2–A 4–E 6–D 8–C 10 – B © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 39 Sample Questions for Physics C: Mechanics Physics C: Mechanics Sample FreeResponse Questions
The following three questions constituted the complete freeresponse section of the 2006
AP Physics C: Mechanics Exam. All freeresponse questions released since 1999 can be
found at AP Central.
Directions: Answer all three questions. The suggested time is about 15 minutes for
answering each of the questions, which are worth 15 points each. The parts within a
question may not have equal weight. Show all your work in the pink booklet in the
spaces provided after each part, NOT in this green insert. Mech 1.
A small block of mass MB = 0.50 kg is placed on a long slab of mass MS = 3.0 kg as
shown above. Initially, the slab is at rest and the block has a speed v0 of 4.0 m/s to the
right. The coefﬁcient of kinetic friction between the block and the slab is 0.20, and there
is no friction between the slab and the horizontal surface on which it moves.
(a) On the dots below that represent the block and the slab, draw and label vectors to
represent the forces acting on each as the block slides on the slab. At some moment later, before the block reaches the right end of the slab, both the block
and the slab attain identical speeds vf .
(b) Calculate vf .
(c) Calculate the distance the slab has traveled at the moment it reaches vf .
(d) Calculate the work done by friction on the slab from the beginning of its motion
until it reaches vf . 40 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Mechanics Mech 2.
A nonlinear spring is compressed various distances x, and the force F required to
compress it is measured for each distance. The data are shown in the table below.
x (m) F (N) 0.05 4 0.10 17 0.15 38 0.20 68 0.25 106 Assume that the magnitude of the force applied by the spring is of the form F(x) = Ax2.
(a) Which quantities should be graphed in order to yield a straight line whose slope
could be used to calculate a numerical value for A ?
(b) Calculate values for any of the quantities identified in (a) that are not given in the
data, and record these values in the table above. Label the top of the column,
including units.
(c) On the axes below, plot the quantities you indicated in (a) . Label the axes with the
variables and appropriate numbers to indicate the scale. (d) Using your graph, calculate A.
The spring is then placed horizontally on the ﬂoor. One end of the spring is ﬁxed to a
wall. A cart of mass 0.50 kg moves on the ﬂoor with negligible friction and collides
headon with the free end of the spring, compressing it a maximum distance of 0.10 m.
(e) Calculate the work done by the cart in compressing the spring 0.10 m from its
equilibrium length.
(f) Calculate the speed of the cart just before it strikes the spring. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 41 Sample Questions for Physics C: Mechanics Mech 3.
A thin hoop of mass M, radius R, and rotational inertia MR 2 is released from rest from
the top of the ramp of length L above. The ramp makes an angle θ with respect to a
horizontal tabletop to which the ramp is ﬁxed. The table is a height H above the ﬂoor.
Assume that the hoop rolls without slipping down the ramp and across the table. Express
all algebraic answers in terms of given quantities and fundamental constants.
(a) Derive an expression for the acceleration of the center of mass of the hoop as it
rolls down the ramp.
(b) Derive an expression for the speed of the center of mass of the hoop when it
reaches the bottom of the ramp.
(c) Derive an expression for the horizontal distance from the edge of the table to where
the hoop lands on the floor.
(d) Suppose that the hoop is now replaced by a disk having the same mass M and
radius R. How will the distance from the edge of the table to where the disk lands
on the floor compare with the distance determined in part (c) for the hoop?
____ Less than ____ The same as ____ Greater than Briefly justify your response. 42 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism Physics C: Electricity and Magnetism
Sample MultipleChoice Questions
Most of the following sample questions have appeared in past exams. The answers are
on page 49. Additional questions can be found in the 2004 AP Physics B and Physics C
Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations listed in the table of information included with the exams (see insert in this book.)
Directions: Each of the questions or incomplete statements below is followed by ﬁve
suggested answers or completions. Select the one that is best in each case. +q
•
–3a 1. x Two charges are located on the xaxis of a coordinate system as shown above. The
charge ϩ2q is located at x = ϩ3a and the charge ϩq is located at x = Ϫ3a. Where
on the xaxis should an additional charge ϩ4q be located to produce an electric
field equal to zero at the origin O?
(A)
(B)
(C)
(D)
(E) 2. O +2q
•
3a x ϭ Ϫ 6a
x ϭ Ϫ 2a
xϭϩa
x ϭ ϩ 2a
x ϭ ϩ 6a A uniform electric field E of magnitude 6,000 V/m exists in a region of space as
shown above. What is the electric potential difference, VX – VY , between points X
and Y ?
(A) –12,000 V
(B)
0V
(C)
1,800 V
(D)
2,400 V
(E)
3,000 V © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 43 Sample Questions for Physics C: Electricity and Magnetism 3. 44 Charge is distributed uniformly throughout a long nonconducting cylinder of
radius R. Which of the following graphs best represents the magnitude of the
resulting electric field E as a function of r, the distance from the axis of the
cylinder? © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism 4. A proton p and an electron e are released simultaneously on opposite sides of an
evacuated area between large, charged parallel plates, as shown above. Each particle
is accelerated toward the oppositely charged plate. The particles are far enough
apart so that they do not affect each other. Which particle has the greater kinetic
energy upon reaching the oppositely charged plate?
(A)
(B)
(C)
(D) The electron
The proton
Neither particle; both kinetic energies are the same.
It cannot be determined without knowing the value of the potential difference
between the plates.
(E) It cannot be determined without knowing the amount of charge on the plates. 5. Two capacitors initially uncharged are connected in series to a battery, as shown
above. What is the charge on the top plate of C1?
(A)
(B)
(C)
(D)
(E) –81 C
–18 C
0 C
+18 C
+81 C © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 45 Sample Questions for Physics C: Electricity and Magnetism b X 6. • b
b •Y Wire of resistivity r and crosssectional area A is formed into an equilateral triangle
of side b, as shown above. The resistance between two vertices of the triangle, X
and Y, is
(A) 3 A
2 rb (B) 3 A
rb
(C) 2 rb
3 A
(D) 3 rb
2 A
(E) 46 3 rb
A © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism Questions 7–8 A particle of electric charge +Q and mass m initially moves along a straight line in the
plane of the page with constant speed v, as shown above. The particle enters a uniform
magnetic ﬁeld of magnitude B directed out of the page and moves in a semicircular arc
of radius R.
7. Which of the following best indicates the magnitude and the direction of the
magnetic force F on the charge just after the charge enters the magnetic field?
Magnitude
kQ2
(A)
R2
kQ2
(B)
R2
(C)
QvB
(D)
QvB
(E)
QvB 8. Direction
Toward the top of the page
Toward the bottom of the page
Out of the plane of the page
Toward the top of the page
Toward the bottom of the page If the magnetic field strength is increased, which of the following will be true about
the radius R?
I. R increases if the incident speed is held constant.
II. For R to remain constant, the incident speed must be increased.
III. For R to remain constant, the incident speed must be decreased.
(A)
(B)
(C)
(D)
(E) I only
II only
III only
I and II only
I and III only © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 47 Sample Questions for Physics C: Electricity and Magnetism 9. 48 A bar magnet is lowered at constant speed through a loop of wire as shown in the
diagram above. The time at which the midpoint of the bar magnet passes through
the loop is t1. Which of the following graphs best represents the time dependence of
the induced current in the loop? (A positive current represents a counterclockwise
current in the loop as viewed from above.) © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism 10. A loop of wire enclosing an area of 1.5 m 2 is placed perpendicular to a magnetic
field. The field is given in teslas as a function of time t in seconds by
B(t) = 20t – 5
3
The induced emf in the loop at t = 3 s is most nearly
(A)
(B)
(C)
(D)
(E) 10 V
15 V
10 V
15 V
20 V Answers to Physics C: Electricity and Magnetism
MultipleChoice Questions
1–A 3–A 5–D 7–E 9–B 2–D 4–C 6–C 8–B 10 – C © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 49 Sample Questions for Physics C: Electricity and Magnetism Physics C: Electricity and Magnetism
Sample FreeResponse Questions
The following three questions constituted the complete freeresponse section of the 2006
AP Physics C: Electricity and Magnetism Exam. All freeresponse questions released
since 1999 can be found at AP Central.
Directions: Answer all three questions. The suggested time is about 15 minutes for
answering each of the questions, which are worth 15 points each. The parts within a
question may not have equal weight. Show all your work in the pink booklet in the
spaces provided after each part, NOT in this green insert. E&M 1.
The square of side a above contains a positive point charge +Q ﬁxed at the lower left
corner and negative point charges –Q ﬁxed at the other three corners of the square. Point
P is located at the center of the square.
(a) On the diagram, indicate with an arrow the direction of the net electric field at
point P.
(b) Derive expressions for each of the following in terms of the given quantities and
fundamental constants.
i. The magnitude of the electric field at point P
ii. The electric potential at point P
(c) A positive charge is placed at point P. It is then moved from point P to point R,
which is at the midpoint of the bottom side of the square. As the charge is moved, is
the work done on it by the electric field positive, negative, or zero?
____ Positive ____ Negative ____ Zero Explain your reasoning.
(d)
i. Describe one way to replace a single charge in this configuration that would
make the electric field at the center of the square equal to zero. Justify your
answer.
ii. Describe one way to replace a single charge in this configuration such that the
electric potential at the center of the square is zero but the electric field is not
zero. Justify your answer.
50 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism E&M 2. The circuit above contains a capacitor of capacitance C, a power supply of emf , two
resistors of resistances R1 and R2, and two switches, S1 and S2. Initially, the capacitor is
uncharged and both switches are open. Switch S1 then gets closed at time t = 0.
(a) Write a differential equation that can be solved to obtain the charge on the capacitor
as a function of time t.
(b) Solve the differential equation in part (a) to determine the charge on the capacitor
as a function of time t.
Numerical values for the components are given as follows: = 12 V C = 0.060 F
R1 = R2 = 4700 Ω
(c) Determine the time at which the capacitor has a voltage 4.0 V across it.
After switch S1 has been closed for a long time, switch S2 gets closed at a new time t = 0.
(d) On the axes below, sketch graphs of the current I1 in R1 versus time and of the
current I2 in R2 versus time, beginning when switch S2 is closed at new time t = 0.
Clearly label which graph is I1 and which is I2. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 51 Sample Questions for Physics C: Electricity and Magnetism E&M 3.
A loop of wire of width w and height h contains a switch and a battery and is connected to a
spring of force constant k, as shown above. The loop carries a current I in a clockwise
direction, and its bottom is in a constant, uniform magnetic ﬁeld directed into the plane of
the page. (a) On the diagram of the loop below, indicate the directions of the magnetic forces, if
any, that act on each side of the loop. (b) The switch S is opened, and the loop eventually comes to rest at a new equilibrium
position that is a distance x from its former position. Derive an expression for the
magnitude B0 of the uniform magnetic field in terms of the given quantities and
fundamental constants. 52 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Sample Questions for Physics C: Electricity and Magnetism The spring and loop are replaced with a loop of the same dimensions and resistance R
but without the battery and switch. The new loop is pulled upward, out of the magnetic
ﬁeld, at constant speed v0. Express algebraic answers to the following questions in terms
of B0, v0, R, and the dimensions of the loop.
(c)
i. On the diagram of the new loop below, indicate the direction of the induced
current in the loop as the loop moves upward. ii. Derive an expression for the magnitude of this current.
(d) Derive an expression for the power dissipated in the loop as the loop is pulled at
constant speed out of the field.
(e) Suppose the magnitude of the magnetic field is increased. Does the external force
required to pull the loop at speed v0 increase, decrease, or remain the same?
_____ Increases _____ Decreases _____ Remains the same Justify your answer. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 53 Teacher Support
AP Central® (apcentral.collegeboard.com)
You can ﬁnd the following Web resources at AP Central:
• AP Course Descriptions, AP Exam questions and scoring guidelines, sample
syllabi, and feature articles.
• A searchable Institutes and Workshops database, providing information about
professional development events.
• The Course Home Pages (apcentral.collegeboard.com/coursehomepages), which
contain articles, teaching tips, activities, lab ideas, and other coursespeciﬁc
content contributed by colleagues in the AP community.
• Moderated electronic discussion groups (EDGs) for each AP course, provided to
facilitate the exchange of ideas and practices. AP Publications and Other Resources
Free AP resources are available to help students, parents, AP Coordinators, and high
school and college faculty learn more about the AP Program and its courses and
exams. Visit www.collegeboard.com/apfreepubs.
Teacher’s Guides and Course Descriptions may be downloaded free of charge from
AP Central; printed copies may be purchased through the College Board Store (store
.collegeboard.com). Released Exams and other priced AP resources are available at
the College Board Store. Teacher’s Guides
For those about to teach an AP course for the ﬁrst time, or for experienced AP teachers
who would like to get some fresh ideas for the classroom, the Teacher’s Guide is an
excellent resource. Each Teacher’s Guide contains syllabi developed by high school
teachers currently teaching the AP course and college faculty who teach the equivalent
course at colleges and universities. Along with detailed course outlines and innovative
teaching tips, you’ll also ﬁnd extensive lists of suggested teaching resources. Course Descriptions
Course Descriptions are available for each AP subject. They provide an outline of each
AP course’s content, explain the kinds of skills students are expected to demonstrate
in the corresponding introductory collegelevel course, and describe the AP Exam.
Sample multiplechoice questions with an answer key and sample freeresponse
questions are included. (The Course Description for AP Computer Science is available
in PDF format only.) Released Exams
Periodically the AP Program releases a complete copy of each exam. In addition to
providing the multiplechoice questions and answers, the publication describes the
process of scoring the freeresponse questions and includes examples of students’
actual responses, the scoring standards, and commentary that explains why the
responses received the scores they did. 54 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. Table of Information and Equation Tables for AP® Physics Exams
The accompanying Table of Information and Equation Tables will be provided to students when
they take the AP Physics Exams. Therefore, students may NOT bring their own copies of these
tables to the exam room, although they may use them throughout the year in their classes in
order to become familiar with their content.
Table of Information
For both the Physics B and Physics C Exams, the Table of Information is printed near the front
cover of the multiplechoice section and on the green insert provided with the freeresponse
section. The tables are identical for both exams except for one convention as noted.
Equation Tables
For both the Physics B and Physics C Exams, the equation tables for each exam are printed only
on the green insert provided with the freeresponse section. The equation tables may be used by
students when taking the freeresponse sections of both exams but NOT when taking the
multiplechoice sections.
The equations in the tables express the relationships that are encountered most frequently in
AP Physics courses and exams. However, the tables do not include all equations that might
possibly be used. For example, they do not include many equations that can be derived by
combining other equations in the tables. Nor do they include equations that are simply special
cases of any that are in the tables. Students are responsible for understanding the physical
principles that underlie each equation and for knowing the conditions for which each equation is
applicable.
The equation tables are grouped in sections according to the major content category in which
they appear. Within each section, the symbols used for the variables in that section are defined.
However, in some cases the same symbol is used to represent different quantities in different
tables. It should be noted that there is no uniform convention among textbooks for the symbols
used in writing equations. The equation tables follow many common conventions, but in some
cases consistency was sacrificed for the sake of clarity.
Some explanations about notation used in the equation tables:
1. The symbols used for physical constants are the same as those in the Table of
Information and are defined in the Table of Information rather than in the righthand
columns of the tables.
2. Symbols in bold face represent vector quantities.
3. Subscripts on symbols in the equations are used to represent special cases of the
variables defined in the righthand columns.
4. The symbol D before a variable in an equation specifically indicates a change in the
variable (i.e., final value minus initial value).
5. Several different symbols (e.g., d, r, s, h, ) are used for linear dimensions such as
length. The particular symbol used in an equation is one that is commonly used for
that equation in textbooks. © 2009 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, and the acorn logo are registered
trademarks of the College Board. TABLE OF INFORMATION FOR 2010 and 2011
CONSTANTS AND CONVERSION FACTORS
27 10 27
31 1.67 Neutron mass, mn 1.67 Electron mass, me
Avogadro’s number, N 0 kg
kg Speed of light,
Universal gravitational
constant,
Acceleration due to gravity
at Earth’s surface, kg 1.38 10 23 1u Planck’s constant, h
hc ⑀0 Vacuum permittivity, Vacuum permeability,
Magnetic constant, k
1 atmosphere pressure, UNIT
SYMBOLS PREFIXES
Factor
Prefix
Symbol 19 J 8 c 3.00 10 m s G 6.67 10 11 m3 kg s2 9.8 m s2 g mole,
hertz,
newton,
pascal,
joule, 10 27 kg 931 MeV c2 6.63 10 34 Js 4.14 1.99 10 25 Jm 1.24 10 12 2 2 1.66 8.85 C Nm 1 4 p⑀0 9.0 4p 10 m0 4 p 1 atm
mol
Hz
N
Pa
J 10 15 eV s 3 10 eV nm 109 N m 2 C2 m0 Coulomb’s law constant, k m
kg
s
A
K C 10 1.60 19 J K 1 unified atomic mass unit, meter,
kilogram,
second,
ampere,
kelvin, 10 1.60 1 electron volt, 1 eV 8.31 J (mol K) Boltzmann’s constant, k B e Electron charge magnitude, 1023 mol 1 6.02 R Universal gas constant, 10 9.11 10 Proton mass, m p 1 10 1.0 7 7 (T m) A (T m) A 105 N m 2 watt,
coulomb,
volt,
ohm,
henry, W
C
V
W
H 1.0 105 Pa farad,
tesla,
degree Celsius,
electronvolt, F
T
C
eV VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
q
30
0
37
45
53
60
90 10 9 giga G sinq 0 12 35 2 2 4 5 3 2 1 106 mega M cosq 1 3 2 4 5 2 2 35 12 0 103 kilo k tanq 0 3 3 34 1 43 3 10 2 centi c 10 3 milli m 10 6 micro m 10 9 nano n 10 12 pico p The following conventions are used in this exam.
I. Unless otherwise stated, the frame of reference of any problem is
assumed to be inertial.
II. The direction of any electric current is the direction of flow of positive
charge (conventional current).
III. For any isolated electric charge, the electric potential is defined as zero at
an infinite distance from the charge.
*IV. For mechanics and thermodynamics equations, W represents the work
done on a system.
*Not on the Table of Information for Physics C, since Thermodynamics is not a
Physics C topic. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. ADVANCED PLACEMENT PHYSICS B EQUATIONS FOR 2010 and 2011
NEWTONIAN MECHANICS u u0 at x x0 u0 t u2 u0 2 1 2
at
2 2a x F
F fric ma Fnet
mN ac u2
r t r F sin q p mv J
K FDt
1 2
mu
2 DUg
W mgh m
N
P
p
r
T
t
U
u
W
x
m
q
t =
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
= acceleration
force
frequency
height
impulse
kinetic energy
spring constant
length
mass
normal force
power
momentum
radius or distance
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque F 1 q1q2
4 p⑀0 r 2 E F
q UE F u cos q Fs V C
C Uc Us 1 2
kx
2 m
k 2p Tp 2p T 1
f FG UG 1
4 p⑀0 g i qi
ri Q
V ⑀0 A
d 1
QV
2 1
CV 2
2 DQ
Dt I avg
R r
A r2 Gm1m2
r e
F
I
P
Q
q
R
r
t
U V =
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
= u =
r =
q =
fm = IV Cp i 1
Cs
Rs 1
Rp Ci 1
i Ci
Ri i i 1
Ri FB qu B sin q FB
Gm1m2 A
B
C
d
E IR kx Ts V
d Eavg P W
Dt 1 q1q2
4 p⑀0 r qV V F Dr cos q Pavg P Dp x0 a
F
f
h
J
K
k ELECTRICITY AND MAGNETISM BI sin q B
fm m0 I
2p r
BA cos q Dfm
Dt eavg e B u © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. area
magnetic field
capacitance
distance
electric field
emf
force
current
length
power
charge
point charge
resistance
distance
time
potential (stored) energy
electric potential or
potential difference
velocity or speed
resistivity
angle
magnetic flux ADVANCED PLACEMENT PHYSICS B EQUATIONS FOR 2010 and 2011
FLUID MECHANICS AND THERMAL PHYSICS WAVES AND OPTICS r mV u fl P P0 n c
u r gh Fbuoy rVg A1u1 A2 u2 P
D 1 2
ru
2 r gy
a const. 0 DT kA DT
L H F
A P PV nRT Nk BT K avg 3
k T
2 B urms 3 RT
M W PDV DU Q 3k B T
m e W W
QH ec TH A = area
e = efficiency
F = force
h = depth
H = rate of heat transfer
k = thermal conductivity
Kavg = average molecular
kinetic energy
= length
L = thickness
m = mass
M = molar mass
n = number of moles
N = number of molecules
P = pressure
Q = heat transferred to a
system
T = temperature
U = internal energy
V = volume
u = velocity or speed
urms = rootmeansquare
velocity
W = work done on a system
y = height
a = coefficient of linear
expansion
m = mass of molecule
r = density TC
TH ATOMIC AND NUCLEAR PHYSICS E hf K max l DE pc
hf f h
p ( Dm) c 2 E=
f =
K=
m=
p =
l=
f= energy
frequency
kinetic energy
mass
momentum
wavelength
work function n 1 sin q1 n 2 sin q2
n2
n1 sin qc
1
si 1
s0 M hi
h0 R
2
d sin q 1
f si
s0 f ml d = separation
f = frequency or
focal length
h = height
L = distance
M = magnification
m = an integer
n = index of
refraction
R = radius of
curvature
s = distance
u = speed
x = position
l = wavelength
q = angle m lL
d xm GEOMETRY AND TRIGONOMETRY Rectangle
A bh
Triangle
1
A
bh
2
Circle
A pr2
C 2p r
Parallelepiped
V
wh
Cylinder
V pr2 A=
C=
V=
S =
b =
h =
=
w=
r = area
circumference
volume
surface area
base
height
length
width
radius S 2p r
2p r 2
Sphere
4 3
V
pr
3
S 4p r 2 Right Triangle
a 2 b2 c2
a
sin q
c
b
cos q
c
a
tan q
b © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. c
q a
90 b ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2010 and 2011
MECHANICS
u u0 at x x0 u0 t u2 u0 2 F 1 2
at
2 2a x x0 ma Fnet
dp
dt F F dt J Dp mv p
F fric mN W F dr K 1 2
mu
2 P dW
dt P F v DUg mgh
2 u
r ac t r
t mr r mr 2 K
w q q0 F
q dV
dr V 1
4 p⑀0 UE Iw C Q
V C k ⑀0 A
d Cp i 1
Cs I Ci 1
Ci
i dQ
dt R r
A E rJ I Neud A V IR Tp
FG at w0 t 1 2
at
2 i 1 q1q2
4 p⑀0 r qV 1 2
kx
2 Ts
p qi
ri 1
QV
2 T 2p
w 1
f UG m
k 2p
2p Rs
g Gm1m2
r2 Gm1m2
r ˆ
r F
I
J
L ⑀0 E P
FM n N = =
=
=
=
=
=
=
V=
u =
r =
fm =
k = P
Q
q
R
r
t
U 1
CV 2
2 area
magnetic field
capacitance
distance
electric field
emf
force
current
current density
inductance
length
number of loops of wire
per unit length
number of charge carriers
per unit volume
power
charge
point charge
resistance
distance
time
potential or stored energy
electric potential
velocity or speed
resistivity
magnetic flux
dielectric constant B dᐉ m0 I dB I dᐉ Bs m0 nI fm e 1
Ri E dᐉ
L UL 1 2
LI
2 IV
qv B B B dA e Ri i m0 I d ᐉ r
4p r3 F i 1
Rp =
=
=
=
=
=
=
=
=
=
=
= A
B
C
d
E e Q E dA Uc 1 2
Iw
2
w0 E kx m rw L 1 q1q2
4 p⑀0 r 2 Us Ia r 2 dm u F Fs F rcm acceleration
force
frequency
height
rotational inertia
impulse
kinetic energy
spring constant
length
angular momentum
mass
normal force
power
momentum
radius or distance
position vector
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque
angular speed
angular acceleration w2 r t net I =
=
=
=
=
=
=
=
=
L =
m=
N =
P =
p =
r =
r =
T =
t =
U=
u =
W=
x =
m =
q =
t =
w =
a = a
F
f
h
I
J
K
k ELECTRICITY AND MAGNETISM © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. dI
dt d fm
dt ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2010 and 2011
GEOMETRY AND TRIGONOMETRY
Rectangle
A bh A=
C=
V=
S =
b =
h =
=
w=
r = Triangle 1
bh
2 A
Circle C 2p r
Parallelepiped
V
wh
Cylinder d f du
du dx d n
x
dx nx n d x
e
dx ex
1
x
cos x d
cos x
dx 2p r n 2 e x dx Sphere 4 3
pr
3 dx
x S 4p r 2 cos x dx a2
sin q b2 1 ln x sin x dx sin x
cos x c2
a
c cos q b
c tan q c
q xn 1, n ex V Right Triangle sin x
1 x n dx 2p r 1 d
1n x
dx pr2 V
S df
dx area
circumference
volume
surface area
base
height
length
width
radius d
sin x
dx pr2 A CALCULUS a
90 b a
b © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. 1 Contact Us
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This note was uploaded on 01/10/2011 for the course PHYS 100 taught by Professor Razasuleman during the Spring '10 term at University of Engineering & Technology.
 Spring '10
 RazaSuleman
 Physics, AP Physics

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