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Assignment MasteringPhysics: Print View
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PHCC 141: Physics for Scientists and Engineers I - Fall 2007
3b. Motion in Two or Three Dimensions (long/challenging start early!)
Due at 11:59pm on Monday, September 10, 2007
Hide Grading Details
Number of answer attempts per question is: 5 You gain credit for: correctly answering a question in a Part, or correctly answering a question in a Hint. You lose credit for: exhausting all attempts or requesting the answer to a question in a Part or Hint, or incorrectly answering a question in a Part. Late submissions: reduce your score by 100% over each day late. Hints are helpful clues or simpler questions that guide you to the answer. Hints are not available for all questions. There is no penalty for leaving questions in Hints unanswered. Grading of Incorrect Answers
For Multiple-Choice or True/False questions, you lose 100%/(# of options - 1) credit per incorrect answer. For any other question, you lose 3% credit per incorrect answer. Standard projectile problems
Shooting over a Hill
A projectile is fired with speed at an angle from the horizontal as shown in the figure .
Part A Find the highest point in the trajectory, . Hint A.1 Velocity at the top Hint not displayed Hint A.2 Which equation to use Hint not displayed Express the highest point in terms of the magnitude of the acceleration due to gravity , the initial velocity , and the angle . ANSWER: =
Part B What is the range of the projectile, ? Part B.1 Find the total time spent in air Part not displayed Part B.2 Find Part not displayed Express the range in terms of , , and . ANSWER: =
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
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Consider your advice to an artillery officer who has the following problem. From his current postition, he must shoot over a hill of height at a target on the other side, which has the same elevation as his gun. He knows from his accurate map both the bearing and the distance to the target and also that the hill is halfway to the target. To shoot as accurately as possible, he wants the projectile to just barely pass above the hill. Part C Find the angle above the horizontal at which the projectile should be fired. Hint C.1 How to approach the problem and in terms of and . Solve these two equations to find in terms of and
In the first half of this problem, you found . Part C.2 Set up the ratio to .
Find the ratio of
The only variable in your answer should be . ANSWER: = and .
Express your answer in terms of ANSWER: =
Recall the following trigonometry formulas: , , and . In this case, since , you can draw a right triangle with as one of the angles, an "opposite" side of length , and an "adjacent" side of length . You can then use this triangle to find and , after you find the length of the hypotenuse using the Pythagorean Theorem.
Part D What is the initial speed? Hint D.1 How to approach this part and . You will need to find an expression for and/or to find . Use one of the equations that you had derived for Part D.2 Find and the Pythagorean theorem to find and .
Use the expression you derived for Leave your answer in terms of ANSWER: =
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Part D.3
Find . and .
Now find the expression for Leave your answer in terms of ANSWER: =
Now use one of the equations that you had derived for hints to find . Express in terms of , , and . =
and
and the expression for
and/or
from the previous
ANSWER:
Part E Find , the flight time of the projectile. Part E.1 First, find ANSWER: How to proceed in terms of and . You can use hints in Part B in the previous half.
= Answer not displayed and .
Express the flight time in terms of ANSWER: =
Projectile and linearly moving object meet
Advice for the Quarterback
A quarterback is set up to throw the football to a receiver who is running with a constant velocity directly away from the quarterback and is now a distance away from the quarterback. The quarterback figures that the ball must be thrown at an angle to the horizontal and he estimates that the receiver must catch the ball a time interval after it is thrown to avoid having opposition players prevent the receiver from making the catch. In the following you may assume that the ball is thrown and caught at the same height above the level playing field. Assume that the y coordinate of the ball at the instant it is thrown or caught is and that the horizontal position of the quaterback is . Use for the magnitude of the acceleration due to gravity, and use the pictured inertial coordinate system when solving the problem.
Part A Find , the vertical component of the velocity of the ball when the quarterback releases it. Part A.1 Equation of motion in y direction What is the expression for , the height of the ball as a function of time? Answer in terms of , , and .
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Part B Find , the initial horizontal component of velocity of the ball. Receiver's position
Part B.1
Find , the receiver's position before he catches the ball. Answer in terms of , , and . ANSWER: Part B.2 = Football's position and .
Find , the horizontal distance that the ball travels before reaching the receiver. Answer in terms of ANSWER: = in terms of , , and .
Express your answer for ANSWER: Part C Find the speed Hint C.1 =
with which the quarterback must throw the ball.
How to approach the problem Hint not displayed
Answer in terms of , , , and . ANSWER: =
Part D Assuming that the quarterback throws the ball with speed , find the angle above the horizontal at which he should throw it. Hint D.1 Find angle from and Hint not displayed Your solution should contain an inverse trig function (entered as asin, acos, or atan). Give your answer in terms of already known quantities, , , and . ANSWER: =
Circular motion and centripetal acceleration NOTE: This next problem is a practice problem - no credit points - but it is recommended to do it since it is a quite detailed tutorial.
Uniform Circular Motion
Learning Goal: To find the velocity and acceleration vectors for uniform circular motion and to recognize that this acceleration is the centripetal acceleration. Suppose that a particle's position is given by the following expression:
. Part A
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Choose the answer that best completes the following sentence: The particle's motion at can be described by ____________. ANSWER: Answer not displayed Part B When does the particle first cross the negative x axis? Express your answer in terms of some or all of the variables , , and . ANSWER: = Answer not displayed
Now, consider the velocity and speed of the particle. Part C Find particle's the velocity as a function of time. Part C.1 Derivative of Part not displayed Express your answer using unit vectors (e.g., ANSWER: Part D Find the speed of the particle at time . Hint D.1 Definition of the magnitude of a vector Hint not displayed Part D.2 Complete an mportant trig identity Part not displayed Express your answer in terms of some or all of the variables , , and . ANSWER: Part E Now find the acceleration of the particle. Express your answer using unit vectors (e.g., ANSWER: Part F Your calculation is actually a derivation of the centripetal acceleration. To see this, express the acceleration of the particle in terms of its position . Express your answer in terms of some or all of the variables ANSWER: Part G Part not displayed Part H = Answer not displayed and . = Answer not displayed + , where and are functions of , , , and ). Answer not displayed = Answer not displayed + , where and are functions of , , , and ).
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MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Finally, express the magnitude of the particle's acceleration in terms of of the particle. Express your answer in terms of one or both of the variables ANSWER: = Answer not displayed
and using the expression you obtained for the speed
and .
Circular Launch
A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 .
Part A How far from the bottom of the chute does the ball land? Part A.1 Speed of ball upon leaving chute How fast is the ball moving at the top of the chute? Hint A.1.a Equation of motion The centripetal acceleration for a particle moving in a circle is rotation. ANSWER: = Time of free fall , where is its speed and is its instantaneous radius of
Part A.2
How long is the ball in free fall before it hits the ground? Part A.2.a Equation of motion There is constant acceleration due to gravity, so you can use the general expression . Write the values of , , and (separated by commas) that are appropriate for this situation. Use the standard convention that is the magnitude of the acceleration due to gravity. Take at the ground, and take the positive y direction to be upward. ANSWER: , , =
Part A.2.b Equation for the height of the ball Part not displayed Express the free-fall time in terms of ANSWER: = and .
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Relative Motion
Crossing a River
A swimmer wants to cross a river, from point A to point B. The distance (from A to C) is 200 , the distance (from C to B) is 150 , and the speed of the current in the river is 5 . Suppose that the swimmer makes an angle of (0.785 ) with respect to the line from A to C, as indicated in the figure.
Part A To swim directly from A to B, what speed , relative to the water, should the swimmer have? Part A.1 Use the motion in the y direction
Suppose it takes the swimmer a time to cross the river and arrive at B. Find an expression for the time by considering only (the y component of the swimmer's motion with respect to the shore). Answer in terms of , , and . ANSWER: = Use the motion in the x direction
Part A.2
Find an expression for by considering only the x component of the swimmer's motion. Part A.2.a Find the x component of velocity relative to shore Find , the x component of the swimmer's velocity relative to the shore. Answer in terms of the variables , , and . ANSWER: =
Give your anwer in terms of , , , and . ANSWER: =
Note that this is the same time ("physically speaking") that you found when you considered only the motion in the y direction, even though the expressions are different. Part A.3 Solve for by eliminating from the two equations you
Find a symbolic expression for . If you've used the previous hints, solve for derived. Express your answer in terms of , , , and . ANSWER: =
Express the swimmer's speed numerically, to three significant figures, in units of kilometers per hour. ANSWER: = 4.04
Another way to do this problem, without using any kinematics, would be to add the swimmer's and river's velocities vectorially, and set the angle that this vector makes with AC or the river bank equal to that which AB makes with the same.
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MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
Problem 3.11
Two crickets, Chirpy and Milada, jump from the top of a vertical cliff. Chirpy jumps horizontally and reaches the ground in a time of 2.10 . Milada jumps with an initial velocity of 90.0 at an angle of 28.0 above the horizontal. Part A How far from the base of the cliff will Milada hit the ground? Take the free fall acceleration to be = 9.80 ANSWER: 170 cm .
Problem 3.18
A rookie quarterback throws a football with an initial upward velocity component of 16.9 component of 20.6 . Ignore air resistance. Part A How much time is required for the football to reach the highest point of the trajectory? Take the free fall acceleration to be = 9.80 ANSWER: Part B How high is this point? Take the free fall acceleration to be = 9.80 ANSWER: Part C How much time (after it is thrown) is required for the football to return to its original level? Take the free fall acceleration to be = 9.80 ANSWER: Part D How far has it traveled horizontally during this time? Take the free fall acceleration to be = 9.80 ANSWER: 71.0 m . 3.45 s . 14.6 m . 1.72 s . and a horizontal velocity
Problem 3.30
A model of a helicopter rotor has four blades, each of length 3.30 in a wind tunnel at a rotational speed of 450 . Part A What is the linear speed of the blade tip? ANSWER: Part B 156 m/s from the central shaft to the blade tip. The model is rotated
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10/4/2007 3:42 PM
MasteringPhysics: Assignment Print View
http://session.masteringphysics.com/myct/assignmentPrint?assignmentI...
What is the radial acceleration of the blade tip expressed as a multiple of the acceleration of gravity, ? ANSWER: 748
Problem 3.37
A "moving sidewalk" in an airport terminal building moves at a speed of 1.00 one end and walks at a speed 1.70 relative to the moving sidewalk. Part A How much time does she require to walk from one end to the other if she walks in the same direction the sidewalk is moving? ANSWER: Part B How much time does she require to walk from one end to the other if she walks opposite to the direction the sidewalk is moving? ANSWER: 57.1 s 14.8 s and is of length 40.0 . A woman steps on at
Problem 3.40
An airplane pilot wishes to fly due west. A wind of 76.0 Part A If the airspeed of the plane (its speed in still air) is 330 , in which direction should the pilot head? is blowing toward the south.
Express your answer as an angle measured north of west ANSWER: Part B What is the speed of the plane over the ground? ANSWER: 321 km/h 13.3 north of west
Summary
9 of 10 items complete (62.86% avg. score) 62.86 of 90 points
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10/4/2007 3:42 PM

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Colorado State - MATH - 161

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CHAPTER 3 DIFFERENTIATION3.1 THE DERIVATIVE OF A FUNCTION 1. Step 1: f(x) oe 4 c x# and f(x b h) oe 4 c (x b h)# Step 2: oe c2x c h Step 3: f w (x) oe lim (c2x c h) oe c2x; f w (c$) oe 6, f w (0) oe 0, f w (1) oe c2h!# # # # # $ # # # #f(x b h)

Colorado State - MATH - 161

CHAPTER 4 APPLICATIONS OF DERIVATIVES4.1 EXTREME VALUES OF FUNCTIONS 1. An absolute minimum at x oe c# , an absolute maximum at x oe b. Theorem 1 guarantees the existence of such extreme values because h is continuous on [a b]. 2. An absolute minimu

Colorado State - MATH - 161

CHAPTER 5 INTEGRATION5.1 ESTIMATING WITH FINITE SUMS 1. faxb oe x# Since f is increasing on ! ", we use left endpoints to obtain lower sums and right endpoints to obtain upper sums.(a) ~x oe (b) ~x oe (c) ~x oe (d) ~x oe 2. faxb oe x$"c! # "c! %

Colorado State - MATH - 161

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS6.1 VOLUMES BY SLICING AND ROTATION ABOUT AN AXIS 1. (a) A oe 1(radius)# and radius oe 1 c x# A(x) oe 1 a1 c x# b (b) A oe width height, width oe height oe 21 c x# A(x) oe 4 a1 c x# b (d) A oe3 4(sid

Colorado State - MATH - 161

CHAPTER 7 TRANSCENDENTAL FUNCTIONS7.1 INVERSE FUNCTIONS AND THEIR DERIVATIVES 1. Yes one-to-one, the graph passes the horizontal test. 2. Not one-to-one, the graph fails the horizontal test. 3. Not one-to-one since (for example) the horizontal line

Colorado State - MATH - 161

CHAPTER 8 TECHNIQUES OF INTEGRATION8.1 BASIC INTEGRATION FORMULAS2.' 3 cos x dx '1 b 3 sin x3.3sin v cos v dv; "4.6.sec z dz tan z 4#z oe1 4du oe cln kukd 1 3 oe ln 3 c ln 1 oe ln 37.'dx x ^ x b 1 u oe x b " " ;

Colorado State - MATH - 161

CHAPTER 9 FURTHER APPLICATIONS OF INTEGRATION9.1 SLOPE FIELDS AND SEPARABLE DIFFERENTIAL EQUATIONS 1. (a) y oe e x y w oe ce x 2y w b 3y oe 2 ace x b b 3e x oe e x (b) y oe e x b e 3x 2 y w oe ce x c 3 e 3x 2 2y w b 3y oe 2 ^ce x c 3 e 3x 2 b 3

Colorado State - MATH - 161

CHAPTER 10 CONIC SECTIONS AND POLAR COORDINATES10.1 CONIC SECTIONS AND QUADRATIC EQUATIONS# # # #1. x oey 8 4p oe 8 p oe 2; focus is (2 0), directrix is x oe c2# #2. x oe c y 4p oe 4 p oe 1; focus is (c1 0), directrix is x oe 1 4 3. y o

Colorado State - MATH - 161

CHAPTER 11 INFINITE SEQUENCES AND SERIES11.1 SEQUENCES 1. a" oe 2. a" oe 3.1 c1 1 1 1!#oe 1, a# oe#" #!oe" 2, a$ oe$1 3!oe1 6, a% oe%1 4!oe" 51 244. a" oe 2 b (c1)" oe 1, a# oe 2 b (c1)# oe 3, a$ oe 2 b (c1)$ oe 1, a%

Colorado State - MATH - 161

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 1. The line through the point (# $ !) parallel to the z-axis 2. The line through the point (c1 0 !) parallel to the y-axis 3. The x-axis 4. The line through the po

Colorado State - MATH - 161

CHAPTER 13 VECTOR-VALUED FUNCTIONS AND MOTION IN SPACE13.1 VECTOR FUNCTIONS 1. x oe t b 1 and y oe t# c 1 y oe (x c 1)# c 1 oe x# c 2x; v oe at t oe 1 2. x oe t# b 1 and y oe 2t c 1 x oe ^ y b 1 b " x oe # v oe i b 2j and a oe 2i at t oe 3. x o

Colorado State - MATH - 161

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