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10:26 MasteringPhysics
4/12/10 AM
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Homework assignment 2 (covers Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6)
Due: 2:00am on Monday, April 12, 2010
Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy
First review Sections 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6 of Young and Freedman, including the worked examples. You should then be able to solve the problems given below. Note that you are allowed only 6 answer attempts per problem.
Given Positions, Find Velocity and Acceleration
Description: Given a table of position versus time, identify graphs of an object's trajectory, velocity, and acceleration. Learning Goal: To understand how to graph position, velocity, and acceleration of an object starting with a table of positions vs. time. The table shows the x coordinate of a moving object. The position is tabulated at 1-s intervals. The x coordinate is indicated below each time. You should make the simplification that the acceleration of the object is bounded and contains no spikes. time (s) x (m) 0 0 1 1 2 4 3 9 4 16 5 24 6 32 7 40 8 46 9 48
Part A Which graph best represents the function object's position vs. time? , describing the
Hint A.1
Meaning of a bounded and nonspiky acceleration vs. .
A bounded and nonspiky acceleration results in a smooth graph of ANSWER:
1 2 3 4
Part B Which of the following graphs best represents the function describing the object's velocity as a function of time? ,
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Hint B.1
Find the velocity toward the end of the motion
Velocity is the time derivative of displacement. Given this, the velocity toward the end of the motion is __________. ANSWER: positive and increasing positive and decreasing negative and increasing negative and decreasing
This narrows the options down to either graph 1 or 4. Hint B.2 What are the implications of zero velocity? and . What would the
Two of the possible velocity vs. time graphs indicate zero velocity between corresponding position vs. time graph look like in this region?
horizontal line straight but sloping up to the right straight but sloping down to the right curved upward curved downward
So graphs 1 and 2 are incorrect. Hint B.3 Specify the characteristics of the velocity function
The problem states that "the acceleration of the object is bounded and contains no spikes." This means that the velocity ___________. ANSWER: has spikes has no discontinuities has no abrupt changes of slope is constant
Which graph or graphs satisfy this condition? ANSWER: 1 2 3 4
In principle, you could also just compute and plot the average velocity. The expression for the average velocity is .
The notation
emphasizes that this is not an instantaneous velocity, but rather an average over an interval.
After you compute this, you must put a single point on the graph of velocity vs. time. The most accurate place to plot the average velocity is at the middle of the time interval over which the average was computed. Also, you could work back and find the position from the velocity graph. The position of an object is the integral of its velocity. That is, the area under the graph of velocity vs. time from up to time must equal the position of the object at time . Check that the correct velocity vs. time graph gives you the correct position according to this method.
Part C
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Which of the following graphs best represents the function describing the acceleration of this object?
,
Hint C.1
Find the acceleration toward the end of the motion
Acceleration is the time derivative of velocity. Toward the end of the motion the acceleration is __________. ANSWER: zero positive negative
Hint C.2
Calculate the acceleration in the region of constant velocity over the interval during which the object travels at constant speed?
What is the acceleration
Answer numerically in meters per second squared. ANSWER: =
Hint C.3
Find the initial acceleration
Acceleration is the time derivative of velocity. Initially the acceleration is _________. ANSWER: zero positive negative
ANSWER:
1 2 3 4
In one dimension, a linear increase or decrease in the velocity of an object over a given time interval implies constant acceleration over that particular time interval. You can find the magnitude of the acceleration using the formula for average acceleration over a time interval: .
When the acceleration is constant over an extended interval, you can choose any value of compute the average.
and
within the interval to
Motion of Two Rockets
Description: Given a stroboscopy image of two rockets (one moving with constant speed, the other with constant acceleration), determine when they have same velocity, position, and acceleration.
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Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated.
Part A At what time(s) do the rockets have the same velocity? Hint A.1 How to determine the velocity
The diagram shows position, not velocity. You can't find instantaneous velocity from this diagram, but you can determine the average velocity between two times and :
.
Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: at time at time at times only only and and
at some instant in time between at no time shown in the figure
Part B At what time(s) do the rockets have the same x position? ANSWER: at time at time at times only only and and
at some instant in time between at no time shown in the figure
Part C At what time(s) do the two rockets have the same acceleration? Hint C.1 How to determine the acceleration
The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next.
ANSWER:
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ANSWER:
at time at time at times
only only and and
at some instant in time between at no time shown in the figure
Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: and nonzero acceleration velocity displacement time
Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: and nonzero acceleration velocity displacement time
Part F At what time(s) is rocket A ahead of rocket B? Hint F.1 Use the diagram
You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: before after before between only only and after and
at no time(s) shown in the figure
One-Dimensional Kinematics with Constant Acceleration
Description: A series of multiple-choice questions about the variables that appear in the standard formulae of one-dimensional kinematics (limited to the case of constant, non-zero acceleration). Learning Goal: To understand the meaning of the variables that appear in the equations for one-dimensional kinematics with constant acceleration. Motion with a constant, nonzero acceleration is not uncommon in the world around us. Falling (or thrown) objects and cars starting and stopping approximate this type of motion. It is also the type of motion most frequently involved in introductory kinematics problems. The kinematic equations for such motion can be written as , , where the symbols are defined as follows: is the position of the particle;
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is the initial position of the particle; is the velocity of the particle; is the initial velocity of the particle; is the acceleration of the particle. In anwering the following questions, assume that the acceleration is constant and nonzero: Part A The quantity represented by false is a function of time (i.e., is not constant). .
Part B The quantity represented by false represents an initial value, not a variable. It refers to the position of an object at some initial moment. is a function of time (i.e., is not constant).
Recall that Part C
The quantity represented by false
is a function of time (i.e., is not constant).
Part D The quantity represented by false always varies with time when the linear acceleration is nonzero. is a function of time (i.e., is not constant).
The velocity
Part E Which of the given equations is not an explicit function of and is therefore useful when you don't know or don't need the time? ANSWER:
Part F A particle moves with constant acceleration . The expression time? ANSWER: at time at the "initial" time when a time has passed since the particle's velocity was represents the particle's velocity at what instant in
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More generally, the equations of motion can be written as
and . Here time and is the time that has elapsed since the beginning of the particle's motion, that is, is the time at which we start measuring the particle's motion. The terms and , where is the current
are, respectively, the position
and velocity at
. As you can now see, the equations given at the beginning of this problem correspond to the case
, which is a convenient choice if there is only one particle of interest. To illustrate the use of these more general equations, consider the motion of two particles, A and B. The position of particle A depends on time as . That is, particle A starts moving at time with velocity , from particle A had at time . At time . , particle B has twice the acceleration, half the velocity, and the same position that
Part G What is the equation describing the position of particle B? Hint G.1 How to approach the problem
The general equation for the distance traveled by particle B is , or , since the equation for ANSWER: is a good choice for B. From the information given, deduce the correct values of the constants that go into given here, in terms of A's constants of motion.
Part H At what time does the velocity of particle B equal that of particle A? Hint H.1 Velocity of particle A
Type an expression for particle A's velocity as a function of time. Hint H.1.1 How to approach this part Look at the general expression for Express your answer in terms of given in the problem introduction. and some or all of the variables , , and .
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ANSWER:
=
Hint H.2
Velocity of particle B
Type an expression for particle B's velocity as a function of time. Hint H.2.1 How to approach this part The general expression for is . From the information given, deduce the correct values of the constants that go into this equation in terms of particle A's constants of motion. Express your answer in terms of ANSWER: and some or all of the variables , , , and .
=
Once you have expressions for the velocities of A and B as functions of time, set them equal and find the time which this happens. ANSWER:
at
The two particles never have the same velocity.
Velocity and Acceleration of a Power Ball
Description: Introduce the concept of a motion diagram, analyze the motion of a power ball and highlight the distinction between velocity and acceleration. Learning Goal: To understand the distinction between velocity and acceleration with the use of motion diagrams. In common usage, velocity and acceleration both can imply having considerable speed. In physics, they are sharply defined concepts that are not at all synonymous. Distinguishing clearly between them is a prerequisite to understanding motion. Moreover, an easy way to study motion is to draw a motion diagram, in which the position of the object in motion is sketched at several equally spaced instants of time, and these sketches (or snapshots) are combined into one single picture. In this problem, we make use of these concepts to study the motion of a power ball. This discussion assumes that we have already agreed on a coordinate system from which to measure the position (also called the position vector) of objects as a function of time. Let and be velocity and acceleration, respectively.
Consider the motion of a power ball that is dropped on the floor and bounces back. In the following questions, you will describe its motion at various points in its fall in terms of its velocity and acceleration. Part A You drop a power ball on the floor. The motion diagram of the ball is sketched in the figure . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
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Hint A.1
Velocity and position vectors
By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the vector position as the distance traveled by the ball, the length of is directly proportional to the length of . Since the length of position vectors is increasing, so is the length of velocity vectors. ANSWER: increasing decreasing not changing
While the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating. Part B Since the length of is directly proportional to the length of , the vector connecting each dot to the next could represent
velocity vectors as well as position vectors, as shown in the figure here . Indicate whether the velocity and acceleration of the ball are, respectively, positive (upward), negative, or zero.
Hint B.1
Acceleration vector
The acceleration is defined as the ratio of the change in velocity to the interval of time, and its direction is given by the quantity , which represents the change in velocity that occurs in the interval of time . Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. ANSWER: N N
Part C Now, consider the motion of the power ball once it bounces upward. Its motion diagram is shown in the figure here . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
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Hint C.1
Velocity and position vectors
By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the vector position as the distance traveled by the ball, the length of is directly proportional to the length of . Since the length of position vectors is decreasing, so is the length of velocity vectors. ANSWER: increasing decreasing not changing
Since the magnitude of the velocity of the ball is decreasing, the ball must be accelerating (specifically, slowing down). Part D The next figure shows the velocity vectors corresponding to the upward motion of the power ball. Indicate whether its velocity and acceleration, respectively, are positive (upward), negative, or zero.
Hint D.1
Acceleration vector
The acceleration is defined as the ratio of the change in velocity to the interval of time, and its direction is given by the quantity , which represents the change in velocity that occurs in the interval of time . Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. ANSWER: P N
Part E The power ball has now reached its highest point above the ground and starts to descend again. The motion diagram representing the velocity vectors is the same as that after the initial release, as shown in the figure of Part B. Indicate whether the velocity and acceleration of the ball at its highest point are positive (upward), negative, or zero. Hint E.1 Velocity as a continuous function of time
In Part D you found that the velocity of the ball is positive during the upward motion. Once the ball starts its descent, its velocity is negative, as you found in Part B. Since velocity changes continuously in time, it has to be zero at some point along the path of the ball. Hint E.2 Acceleration as a continuous function of time
In Part D, you found that the acceleration of the ball is negative and constant during the upward motion, as well as once the ball has started its descent, which you found in Part B. Since acceleration is a continuous function of time, it has to be negative at the highest point along the path as well. Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. ANSWER: Z N
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These examples should show you that the velocity and acceleration can have opposite or similar signs or that one of them can be zero while the other has either sign. Try hard to think carefully about them as distinct physical quantities when working with kinematics.
What Velocity vs. Time Graphs Can Tell You
Description: The relation between area under the v vs. t curve and distance traveled is presented. Find average and instantaneous acceleration given a v vs. t graph. A common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time is plotted on the horizontal axis and velocity on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only a single nonzero component in the direction of motion. Thus, in this problem, we will call the velocity and the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion, respectively. Here is a plot of velocity versus time for a particle that travels along a straight line with a varying velocity. Refer to this plot to answer the following questions.
Part A What is the initial velocity of the particle, Hint A.1 Initial velocity . ?
The initial velocity is the velocity at Hint A.2 How to read a v vs. t graph
Recall that in a graph of velocity versus time, time is plotted on the horizontal axis and velocity on the vertical axis. For example, in the plot shown in the figure, at . Express your answer in meters per second. ANSWER: =
Part B What is the total distance Hint B.1 traveled by the particle?
How to approach the problem under the v vs. t curve is always equal to the distance
Recall that the area of the region that extends over a time interval traveled in
. Thus, to calculate the total distance, you need to find the area of the entire region under the v vs. t curve. In
the case at hand, the entire region under the v vs. t curve is not an elementary geometrical figure, but rather a combination of triangles and rectangles. Hint B.2 Find the distance traveled in the first 20.0 seconds traveled in the first 20 seconds of motion, between and ?
What is the distance
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Hint B.2.1 Area of the region under the v vs. t curve The region under the v vs. t curve between by , and a triangle of base and and height can be divided into a rectangle of dimensions , as shown in the figure.
Express your answer in meters. ANSWER: =
Hint B.3
Find the distance traveled in the second 20.0 seconds traveled in the second 20 seconds of motion, from to ?
What is the distance
Hint B.3.1 Area of the region under the v vs. t curve The region under the v vs. t curve between as shown in the figure. and is a rectangle of dimensions by ,
Express your answer in meters. ANSWER: =
Hint B.4
Find the distance traveled in the last 10.0 seconds traveled in the last 10 seconds of motion, from to ?
What is the distance
Hint B.4.1 Area of the region under the v vs. t curve The region under the v vs. t curve between as shown in the figure. and is a triangle of base and height ,
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Express your answer in meters. ANSWER: =
Now simply add the distances traveled in each time interval to find the total distance. Express your answer in meters. ANSWER: =
Part C What is the average acceleration Hint C.1 of the particle over the first 20.0 seconds?
Definition and graphical interpretation of average acceleration of a particle that travels along a straight line in a time interval , or . is the ratio of the change in
The average acceleration velocity
experienced by the particle to the time interval
In a v vs. t graph, then, the average acceleration equals the slope of the line connecting the two points representing the initial and final velocities. Hint C.2 The slope the "run," or . Slope of a line of a line from point A, of coordinates , to point B, of coordinates , is equal to the "rise" over
Express your answer in meters per second per second. ANSWER: =
The average acceleration of a particle between two instants of time is the slope of the line connecting the two corresponding points in a v vs. t graph. Part D What is the instantaneous acceleration Hint D.1 of the particle at ?
Graphical interpretation of instantaneous acceleration
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The acceleration of a particle at any given instant of time or at any point in its path is called the instantaneous acceleration. If the v vs. t graph of the particle's motion is known, you can directly determine the instantaneous acceleration at any point on the curve. The instantaneous acceleration at any point is equal to the slope of the line tangent to the curve at that point. Hint D.2 The slope the "run," or . Slope of a line of a line from point A, of coordinates , to point B, of coordinates , is equal to the "rise" over
ANSWER:
1 0.20 = -0.20 0.022 -0.022
The instantaneous acceleration of a particle at any point on a v vs. t graph is the slope of the line tangent to the curve at that point. Since in the last 10 seconds of motion, between and , the curve is a straight line, the tangent line is the curve itself. Physically, this means that the instantaneous acceleration of the particle is constant over that time interval. This is true for any motion where velocity increases linearly with time. In the case at hand, can you think of another time interval in which the acceleration of the particle is constant? Now that you have reviewed how to plot variables as a function of time, you can use the same technique and draw an acceleration vs. time graph, that is, the graph of (instantaneous) acceleration as a function of time. As usual in these types of graphs, time is plotted on the horizontal axis, while the vertical axis is used to indicate acceleration .
Part E Which of the graphs shown below is the correct acceleration vs. time plot for the motion described in the previous parts? Hint E.1 How to approach the problem
Recall that whenever velocity increases linearly with time, acceleration is constant. In the example here, the particle's velocity increases linearly with time in the first 20.0 of motion. In the second 20.0 , the particle's velocity is constant, and then it decreases linearly with time in the last 10 . This means that the particle's acceleration is constant over each time interval, but its value is different in each interval. Hint E.2 What is Find the acceleration in the first 20 , the particle's acceleration in the first 20 of motion, between and ?
Hint E.2.1 Constant acceleration Since we have already determined that in the first 20 of motion the particle's acceleration is constant, its constant value will be equal to the average acceleration that you calculated in Part C. Express your answer in meters per second per second. ANSWER: =
Hint E.3 What is
Find the acceleration in the second 20 , the particle's acceleration in the second 20 of motion, between and ?
Hint E.3.1 Constant velocity In the second 20 of motion, the particle's velocity remains unchanged. This means that in this time interval, the particle does not accelerate. Express your answer in meters per second per second.
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ANSWER:
=
Hint E.4 What is
Find the acceleration in the last 10 , the particle's acceleration in the last 10 of motion, between and ?
Hint E.4.1 Constant acceleration Since we have already determined that in the last 10 of motion the particle's acceleration is constant, its constant value will be equal to the instantaneous acceleration that you calculated in Part D. Express your answer in meters per second per second. ANSWER: =
ANSWER:
Graph A Graph B Graph C Graph D
In conclusion, graphs of velocity as a function of time are a useful representation of straight-line motion. If read correctly, they can provide you with all the information you need to study the motion.
What x vs. t Graphs Can Tell You
Description: Find average and instantaneous velocity given an x vs. t graph. The relation between area under the v vs. t curve and distance traveled is presented. To describe the motion of a particle along a straight line, it is often convenient to draw a graph representing the position of the particle at different times. This type of graph is usually referred to as an x vs. t graph. To draw such a graph, choose an axis system in which time is plotted on the horizontal axis and position on the vertical axis. Then, indicate the values of at various times . Mathematically, this corresponds to plotting the variable as a function of . An example of a graph of position as a function of time for a particle traveling along a straight line is shown below. Note that an x vs. t graph like this does not represent the path of the particle in space. Now let's study the graph shown in the figure in more detail.
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Refer to this graph to answer Parts A, B, and C.
Part A What is the total distance Hint A.1 Total distance traveled by the particle is given by the difference between the initial position . In symbols, . Hint A.2 How to read an x vs. t graph is plotted on the horizontal axis and position at . on the vertical axis. For example, at and the traveled by the particle?
The total distance position at
Remember that in an x vs. t graph, time in the plot shown in the figure, Express your answer in meters. ANSWER: =
Part B What is the average velocity Hint B.1 of the particle over the time interval ?
Definition and graphical interpretation of average velocity of a particle that travels a distance along a straight line in a time interval . is defined as
The average velocity
In an x vs. t graph, then, the average velocity equals the slope of the line connecting the initial and final positions. Hint B.2 The slope Slope of a line of a line from point A, with coordinates , to point B, with coordinates , is equal to the "rise"
over the "run," or .
Express your answer in meters per second. ANSWER: =
The average velocity of a particle between two positions is equal to the slope of the line connecting the two corresponding points in an x vs. t graph.
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Part C What is the instantaneous velocity Hint C.1 of the particle at ?
Graphical interpretation of instantaneous velocity
The velocity of a particle at any given instant of time or at any point in its path is called instantaneous velocity. In an x vs. t graph of the particle's motion, you can determine the instantaneous velocity of the particle at any point in the curve. The instantaneous velocity at any point is equal to the slope of the line tangent to the curve at that point. Express your answer in meters per second. ANSWER: =
The instantaneous velocity of a particle at any point on its x vs. t graph is the slope of the line tangent to the curve at that point. Since in the case at hand the curve is a straight line, the tangent line is the curve itself. Physically, this means that the instantaneous velocity of the particle is constant over the entire time interval of motion. This is true for any motion where distance increases linearly with time. Another common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time is plotted on the horizontal axis and velocity on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only one nonzero component in the direction of motion. Thus, in this problem, we will call the velocity and the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion. Part D Which of the graphs shown is the correct v vs. t plot for the motion described in the previous parts? Hint D.1 How to approach the problem
Recall your results found in the previous parts, namely the fact that the instantaneous velocity of the particle is constant. Which graph represents a variable that always has the same constant value at any time?
ANSWER:
Graph A Graph B Graph C Graph D
Whenever a particle moves with constant nonzero velocity, its x vs. t graph is a straight line with a nonzero slope, and its v vs. t curve is a horizontal line.
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Part E Shown in the figure is the v vs. t curve selected in the previous part. What is the area of the shaded region under the curve?
Hint E.1
How to approach the problem axis,
The shaded region under the v vs. t curve is a rectangle whose horizontal and vertical sides lie on the axis and the
respectively. Since the area of a rectangle is the product of its sides, in this case the area of the shaded region is the product of a certain quantity expressed in seconds and another quantity expressed in meters per second. The area itself, then, will be in meters. Express your answer in meters. ANSWER: =
Compare this result with what you found in Part A. As you can see, the area of the region under the v vs. t curve equals the total distance traveled by the particle. This is true for any velocity curve and any time interval: The area of the region that extends over a time interval under the v vs. t curve is always equal to the distance traveled in .
Overcoming a Head Start
Description: Two cars start at different positions and travel at different, but constant, speeds. Determine when and where the trailing car catches up with the lead car. Cars A and B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance beyond the starting line at . The starting line is at . Car A travels at a constant speed . Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed Part A How long after Car B started the race will Car B catch up with Car A? Hint A.1 Consider the kinematics relation after Car B starts. (Note that we are .) , which is greater than .
Write an expression for the displacement of Car A from the starting line at a time taking this time to be
Hint A.1.1 What is the acceleration of Car A? The acceleration of Car A is zero, so the general formula Answer in terms of ANSWER: , , , and for time, and take at the starting line. has at least one term equal to zero.
=
Hint A.2
What is the relation between the positions of the two cars?
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The positions of the two cars are equal at time Hint A.3
.
Consider Car B's position as a function of time after starting. for time).
Write down an expression for the position of Car B at time Give your answer in terms of any variables needed (use ANSWER: =
Express the time in terms of given quantities. ANSWER: =
Part B How far from Car B's starting line will the cars be when Car B passes Car A? Hint B.1 Which expression should you use? , and substitute in the correct value for (found in
Just use your expression for the position of either car after time the previous part). Express your answer in terms of known quantities. (You may use ANSWER:
as well.)
=
Motion of a Shadow
Description: This problem provides practice with basic geometry and velocity using the example of the motion of a shadow on a wall relative to motion of an obstacle. Involves simple differentiation. A small source of light is located at a distance from a vertical
wall. An opaque object with a height of with constant velocity
moves toward the wall , the
of magnitude . At time .
object is located at the source
Part A Find an expression for Hint A.1 , the magnitude of the velocity of the top of the object's shadow, at time .
Calculate ratios of triangles
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Triangles
and
are similar, so the ratios of the
corresponding sides are equal. Hence, the ratio and . Therefore, solve for the length in terms of , , and .
ANSWER: =
Note that instead of referring to similar triangles, we could have used the fact that angle triangles: ,
is the same in both
which leads quickly to the ratios of the lengths of corresponding sides being equal. Hint A.2 Calculate the derivative . To find the shadow's velocity, ?
You have an expressin for the position of the object's shadow as a function of time, take the derivative of Hint A.2.1 The chain rule You may need to use the chain rule. Recall: . Note that and are constants. What is
Leave your answer in terms of ANSWER: =
and
.
Hint A.3
Find should contain . You need to substitute this with a variable given in the problem.
Your expression for Hint A.3.1 Meaning of
is a quantity that tells you how much
changes as
changes. It is the rate of change of
. Look for the variable
that tells you how quickly or slowly the distance from the object to the light changes. Which of the quantities given in the problem statement is represented by ANSWER: ?
=
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Hint A.4 Find
Substitute , the position of the object as a function of time in terms of the quantities given in the problem.
ANSWER:
=
Express the speed of the top of the object's shadow in terms of , , ANSWER: =
, and
.
Rearending Drag Racer
Description: The problem examines acceleration, distance and velocity through the example of a car catching an accelerating dragster which is given a headstart. To demonstrate the tremendous acceleration of a top fuel drag racer, you attempt to run your car into the back of a dragster that is "burning out" at the red light before the start of a race. (Burning out means spinning the tires at high speed to heat the tread and make the rubber sticky.) You drive at a constant speed of toward the stopped dragster, not slowing down in the face of the imminent collision. The dragster driver sees you coming but waits until the last instant to put down the hammer, accelerating from the starting line at constant acceleration, . Let the time at which the dragster starts to accelerate be .
Part A What is , the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity? Hint A.1 At Calculate the velocity
, what will the velocity of the drag car be?
Hint A.1.1 Consider the speed of both cars No collision can occur if the dragster has greater speed than the speed of the car behind it. Your answer should not contain tmax, as that time is not yet known. ANSWER: =
ANSWER: =
Part B Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the dragster at ), find your distance from the dragster when he started. If you calculate positions on the way to this solution, choose coordinates so that the position of the drag car is 0 at . Remember that you are solving for a distance (which is a magnitude, and can never be negative), not a position (which can be negative).
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Hint B.1 Taking
Drag car position at time at the position of the dragster at , find , the position of the dragster at .
Express your answer in terms of ANSWER:
and given quantities.
=
.
Hint B.2 Find
Distance car travels until tmax , the distance you travel from to .
ANSWER: =
Hint B.3 Express
Starting position of car , the distance the car travels in terms of the starting distance of the car from the starting line at time , . Note that ,
, and the position of the drag car at time should affect your use of signs. Express your answer in terms of ANSWER: ,
is a distance and can't be negative. This
, and
.
=
Hint B.4 solve for ANSWER:
Obtaining the Solution , substitute for in terms of and , and and . , the initial distance of the car from the starting line. Your answer should be in terms of
Equate your two expressions for the distance traveled by the car up to
=
Part C Find numerical values for (26.8 m/s) and and in seconds and meters for the (reasonable) values .
Separate your two numerical answers by commas, and give your answer to two significant figures.
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ANSWER: , = s, m
The blue curve shows how the car, initially at accelerating drag car (red) at .
, continues at constant velocity (blue) and just barely touches the
Rocket Height
Description: This problem looks at constant acceleration and velocity using the example of a rocket. No air resistance is involved. A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 29.4 acceleration period lasts for time 9.00 Part A Find the maximum height equal to 9.8 Hint A.1 . How to approach the problem reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity until the fuel is exhausted. After that, the rocket is in free fall. . The
Divide the upward motion into two parts: first the fueled motion, and then the motion under the influence of gravity alone. Find the height reached over the course of the fueled motion, and then calculate the additional height achieved during the second part of the motion. Putting these two distances together will give you the maximum height reached by the rocket. Hint A.2 Find the height reached during the fueled part of the motion above the ground at which the rocket exhausts its fuel.
Find the height
Hint A.2.1 Knowns and unknowns At the instant that the rocket takes off, take time and the initial position . Let the final values of the for the fueled portion of the
variables correspond to those at which the rocket runs out of fuel. Clearly, the final height flight and the associated final velocity are not given. Let us denote other quantities as follows: rocket's initial velocity; and quantities are known?
is the time that the rocket travels before it runs out of fuel;
is the
is the rocket's net acceleration during the fueled portion of its flight. Which of the these
Hint A.2.1.1 What is the initial velocity? What is the initial velocity Give your answer numerically. ANSWER: = for the fueled part of the motion?
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Note: For the fueled part of the motion the variable called usually is being called the initial velocity for the second part of the motion, which we will call . Check all that apply. ANSWER:
, to distinguish it from
Note: For the fueled part of the motion the variables usually labeled
and
are labeled
and
, to
distinguish them from the initial velocity and acceleration for the second part of the motion, which we will call and respectively. Hint A.2.2 Determine which kinematic equation to use Choose the kinematic equation that makes the solution straighforward, that is, the one that contains the variable you are solving for and in which all of the other quantities are known. ANSWER:
Now substitute the given values into this equation to find the height Answer numerically in units of meters. ANSWER: =
.
Note that the upward acceleration of the rocket results from both the thrust of the engine and from the force due to gravity; thus, the existence of gravity is already "taken into account" in the statement of the problem. You can now either find the total height that the rocket reaches or first determine the additional vertical distance the rocket travels after it runs out of fuel and add this value to the value you found for . Since you don't know the time it takes for the rocket to reach its maximum height, you must determine the quantities that you do know for this part of the motion: the initial velocity , the final velocity , and the acceleration . Look at the figure for a clearer picture.
Hint A.3 What are
Find the initial velocity, the final velocity, and the acceleration for the "free-fall" part of the motion , , and for the second part of the motion?
Hint A.3.1 What is the initial velocity? When the rocket runs out of fuel, its acceleration changes abruptly, but its velocity changes continuously. Therefore, the
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rocket's initial velocity What, then, is ?
for the second part of the flight is just its velocity at the moment the engine runs out of fuel.
Hint A.3.1.1 Find the velocity when the engine runs out of fuel For the fueled part of the motion, you know that the initial velocity is given by 29.4 , and the time of fueled flight by , the acceleration by in Part A.2.
9.00 . You also determined the height
Choose a kinematic equation that you could use to find A. B. C. D.
, the velocity at the end of the fueled motion.
Choose one letter corresponding to the equation you have chosen (even though there is more than one correct answer). B C D A B C D A B C D Now substitute the given values into this equation to find Give your answer numerically. ANSWER: = , which is equal to .
Hint A.3.2 What is the acceleration? What value should you use for the acceleration ? Keep in mind that the direction is important, since the acceleration due to gravity is slowing down the rocket as it continues its ascent. Give your answer numerically. ANSWER: =
Hint A.3.3 What is the final velocity? What is the velocity of the rocket when it reaches its maximum height? Note that the rocket has just ended its ascent and is about to begin its descent. What is its velocity at this instant? Give your answer numerically. ANSWER: =
Write your answer numerically in the order ANSWER:
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, separated by commas as shown, in SI units.
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ANSWER: = SI units
Look at the figure below for a nice way to represent all this data.
Hint A.4 Determine which kinematic equation to use Choose the kinematic equation that makes the solution straightforward, that is, the one that contains the variable you are solving for and for which all of the other quantities are known. ANSWER:
Now substitute the given values into this equation to find either the total height additional height gained (if you use ). Write your answer numerically in units of meters. ANSWER: =
(if you use
) or the
Basketball Jump Shot
Description: A basketball player jumps to take a shot. An identical player jumps a short time later to block the shot. Determine when the first player has the maximum height above the second player (and therefore has the best shot). Two basketball players are essentially equal in all respects. (They are the same height, they jump with the same initial velocity, etc.) In particular, by jumping they can raise their centers of mass the same vertical distance, (called their "vertical leap"). The first player, Arabella, wishes to shoot over the second player, Boris, and for this she needs to be as high above Boris as possible. Arabella jumps at time , and Boris jumps later, at time (his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps. Part A Find the vertical displacement height of the raised hands of Arabella, while Hint A.1 How to approach the problem , as a function of time for the interval is the height of the raised hands of Boris. , where is the
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This problem is concerned with the distance between the players' hands before the second player, Boris, has jumped. Thus, you need to consider only the motion of the first player, Arabella. During this time interval, Boris stays on the ground. The trickiest part of this problem is to determine Arabella's initial velocity in terms of the information given. Hint A.2 Type of motion of Arabella's jump
Arabella's vertical motion is motion with constant _____ . ANSWER:
acceleration
Hint A.3
Find Arabella's position as a function of time , Arabella jumps from a height with initial velocity . If the only force acting on Arabella
Assume that at time
after she jumps is gravity, find a general expression for her height Express your answer in terms of ANSWER: =
as a function of time.
, , and , the magnitude of the acceleration due to gravity.
Since Boris has not jumped yet in this part, the result you just obtained gives you the height of Arabella above Boris. However, you need to eliminate the unknown initial velocity . Hint A.4 Find Find Arabella's initial velocity ).
, Arabella's initial velocity (at time
Hint A.4.1 How to approach the problem Write an equation for Arabella's velocity, at which at the time , as a function of time. Arabella's maximum height occurs at the time
. Solving this equation will give you the time it takes for Arabella to reach her maximum height, . Finally, evaluate the general equation for Arabella's position as a function of time at this time.) Solve the resulting expression to find in . (You know that Arabella's height will be
, in terms of her initial velocity terms of other given quantities. Hint A.4.2 Find
How long after jumping does it take Arabella to reach her maximum height? Hint A.4.2.1 How to approach the question Use the equation along with the fact that . Remember that the acceleration due to gravity
(magnitude ) is directed opposite to Arabella's initial velocity. Express your answer in terms of ANSWER: = and .
Hint A.4.3 Plug
into Arabella's equation of motion , Arabella is at height . In other words, by substituting into the ,
From the problem introduction, you know that at time where
gives Arabella's height as a function of time. Find an expression for (found in Hint 3). , , and .
general expression for
Express your answer in terms of
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ANSWER:
=
Rearrange the equation you just obtained to find Express your answer in terms of the given constants ANSWER: =
in terms of and .
and .
Express the vertical displacement in terms of ANSWER: =
, , and .
Part B Find the vertical displacement between the raised hands of the two players for the time period after Boris has jumped (
) but before Arabella has landed. Hint B.1 How to approach the problem
In Part A you found Arabella's height as a function of time. Now find Boris's height as a function of time. Subtract Boris's height from Arabella's height to find the difference in height. Hint B.2 Find Find Boris's height , Boris's height above the ground, for .
Hint B.2.1 Use a change of variables Since Arabella and Boris are equal in all respects as basketball players, the only difference between Arabella's jump and Boris's jump is that Boris starts at time instead of . Another way of saying this is that at any moment in time while both players are in the air, Boris's height is equal to whatever Arabella's height was Mathematically, this means that ANSWER: , where is related to by _______. time units earlier.
=
Express your answer in terms of , ANSWER: =
,
, and .
Express your answer in terms of , ANSWER: =
, , and
.
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Part C What advice would you give Arabella to minimize the chance of her shot being blocked? ANSWER: Shoot when you have the maximum vertical velocity. Shoot at the instant Boris leaves the ground. Shoot when you have the same vertical velocity as Boris. Shoot when you reach the top of your jump (when your height is
).
A Flower Pot Falling Past a Window
Description: A flower pot falls with constant acceleration from some unknown height. Given information from the middle of its motion (as it falls past a window) determine the height from which it was dropped and the speed it will have when it lands. As you look out of your dorm window, a flower pot suddenly falls past. The pot is visible for a time , and the vertical length of your window is . Take down to be the positive direction, so that downward velocities are positive and the acceleration due to gravity is the positive quantity . Assume that the flower pot was dropped by someone on the floor above you (rather than thrown downward). Part A From what height Hint A.1 above the bottom of your window was the flower pot dropped?
How to approach the problem
The initial velocity of the pot is zero. Find the velocity of the pot at the bottom of the window. Then using the kinematic equation that relates initial and final velocities, acceleration, and distance traveled, you can solve for the distance . Hint A.2 Find the velocity at the bottom of the window What is the velocity of the flower pot at the instant it passes the bottom of your window?
Hint A.2.1 Find the average velocity What is the average velocity of the flower pot as it passes by your window? and .
Express your answer in terms of ANSWER: =
Now, think about when the pot actually has this velocity Hint A.2.2 Find the time when
.
As the pot falls past your window, there will be some instant when the pot's velocity equals the average velocity . How much time does it take, after the pot's instantaneous velocity equals its average velocity, for the pot to reach the bottom of the window? Recall that, under constant acceleration, velocity changes linearly with time. This means that the average velocity during a time interval will occur at the middle of that time interval. Express your answer in terms of . ANSWER: =
Now combine this with the acceleration to find the difference between gives . Express your answer in terms of , , and .
and
. Adding this difference to
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ANSWER: =
Hint A.3 The needed kinematic equation To solve this problem most easily, you should use the kinematic equation looking for . Note that you are
, the distance traveled by the flower pot from the moment it was dropped until it reaches the height of the
bottom of your window. Express your answer in terms of ANSWER: = , , and .
Part B If the bottom of your window is a height may introduce the new variable above the ground, what is the velocity of the pot as it hits the ground? You
, the speed at the bottom of the window, defined by .
Hint B.1
Needed kinematic equation . Using the kinematic equation that relates initial and
The initial velocity of the pot is zero. The total distance it falls is
final velocities, the total distance traveled, and the acceleration, you can solve for the pot's final velocity. Alternatively, you could use the same kinematic equation, but set Hint B.2 Find the initial height of the pot above the ground was the pot dropped? , , , , and . and .
From what height
Express your answer in terms of some or all of the variables ANSWER:
=
Express your answer in terms of some or all of the variables ANSWER:
,
, ,
, and .
=
Going for a Drive
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Description: Qualitative aspects of constant acceleration kinematics followed by quantitative questions. Uses applets. Learning Goal: To gain a qualitative understanding of kinematics and how the qualitative nature of position and velocity versus time graphs relates to the equations of kinematics. In this problem, you will explore kinematics using an applet that simulates a car moving under constant acceleration. When you open the applet, you will see three sliders that allow you to adjust the initial position , the initial velocity , and the acceleration . Set the initial position to 0 , the initial velocity to and the acceleration to 5 .
Run the simulation. Notice that, as the movie proceeds, pictures of the car remain at certain points. Once the simulation is over, these pictures form a motion diagram--a representation of motion consisting of pictures taken at equal time intervals during the motion. In this case, the interval between pictures is one second. Below the movie, the position of the car as a function of time is graphed in green. Run the simulation several times, paying attention to how the graph and the motion diagram/movie of the car's motion relate to each other. Part A Which of the following describe the relationship between the motion diagram/movie and the graph? Check all that apply. ANSWER: When the slope of the graph is close to zero, the pictures in the motion diagram are close together. When the slope of the graph is steep, the car is moving quickly. When the slope of the graph is positive, the car is to the right of its starting position. When the x position on the graph is negative, the car moves backward. When the x position on the graph is positive, the car moves forward. When the x position on the graph is negative, the car moves slowly. When the x position on the graph is positive, the car moves quickly. When the x position on the graph is negative, the car is to the left of its starting position. When the x position on the graph is positive, the car is to the right of its starting position. , , and . The last option,
Notice that the first and second options are always true, regardless of the values of however, is only true when is often a good choice.
. Frequently, you will be able to pick your coordinate system. In such cases, making
Part B Run the simulation, paying close attention to the graph of position. Press reset and change the value of again, noting any changes in the graph. How does varying affect the graph of position? ANSWER: Increasing Increasing Increasing Increasing Increasing Changing . Run the simulation
increases the width of the graph, whereas decreasing decreases the width. shifts the graph to the right, whereas decreasing it shifts the graph to the left. shifts the graph to the left, whereas decreasing it shifts the graph to the right. shifts the graph upward, whereas decreasing it shifts the graph downward. shifts the graph downward, whereas decreasing it shifts the graph upward. does not affect the graph.
Part C Now, run the simulation with different values of does varying affect the graph of position? Choose the best answer. ANSWER: Increasing increases the width of the graph, whereas decreasing decreases the width. Increasing shifts the graph to the right and upward, whereas decreasing it shifts the graph to the left and downward. Increasing shifts the graph to the left and upward, whereas decreasing it shifts the graph to the right and downward. Increasing shifts the graph to the right and downward, whereas decreasing it shifts the graph to the left and upward. Increasing shifts the graph to the left and downward, whereas decreasing it shifts the graph to the right and upward. Changing does not affect the graph. , but don't use any positive values. Note any changes in the graph. How
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This behavior may be a bit difficult to understand by just looking at the equation for position vs. time that you know from kinematics. If you complete the square to get the equation into standard form for a parabola, this should become more apparent. Now that you've seen how affects the graph, run the simulation with a few different values of acceleration. You should see that increasing the acceleration decreases the width of the graph and decreasing the acceleration increases the width. (Decreasing the acceleration below 0 makes the parabola open downward instead of upward.)
Part D Enter the equation for position as a function of time. Before submitting your answer, check that it is consistent with the in the equation, would it move the graph , and acceleration .
qualities of the graph that you have identified. For instance, if you increase upward? Express your answer in terms of time , initial position ANSWER: = , initial velocity
Part E Now, open this applet. This applet looks like the previous applet, but when you run the simulation, you will now get graphs of both position and velocity. Run the simulation several times with different values of . How does changing affect the graph of velocity? ANSWER: Increasing Increasing Increasing Increasing Increasing Changing increases the slope of the graph, whereas decreasing decreases the slope. shifts the graph to the right, whereas decreasing it shifts the graph to the left. shifts the graph to the left, whereas decreasing it shifts the graph to the right. shifts the graph upward, whereas decreasing it shifts the graph downward. shifts the graph downward, whereas decreasing it shifts the graph upward. does not affect the graph.
Part F Run the simulation again, with the following settings: the graph are seconds. At what time , , and . The units of time in
is the velocity equal to zero?
Express your answer in seconds to the nearest integer. ANSWER:
Notice that the position graph has a minimum when velocity equals zero. This should make sense to you. Since velocity is the derivative of position, position has a local minimum or maximum when velocity is zero. Part G Suppose that a car starts from rest at position time is the velocity of the car 19.2 and accelerates with a constant acceleration of 4.15 . At what
? Use the applet to be certain that your answer is reasonable.
Hint G.1
Choose the kinematic equation
You have seen a number of kinematic equations. Choose the one from the following list that will be the most useful in this problem. Again, is the acceleration, and are, respectively, velocity and position at time , while and are, respectively, the initial velocity and initial position. ANSWER:
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Hint G.2
Using the applet to check your answer
The applet does not allow you to enter numbers to the precision asked for in the problem. However, you can check that your answer is reasonable by selecting values close to the ones asked for in the problem. For instance, you cannot use the acceleration 4.15 with the applet, but you can use 4 . If you set all of the values on the applet as close as you can to the values in the problem, then the answer seen on the applet should be close to the correct answer. If the applet shows an answer of 10 and your calculations give you 5 , then you should check your calculations for errors and be sure that you're using the right equations and principles. However, if your calculations give 9.5 , there is a good chance that you are right. Express your answer in seconds to three significant figures. ANSWER: =
Part H For the same initial conditions as in the last part, what is the car's position check that your answer is reasonable. Hint H.1 Choose the kinematic equation at time 4.05 ? Again, be sure to use the applet to
You have seen a number of kinematic equations. Choose the one from the following list that will be the most useful in this problem. Again, is the acceleration, and are, respectively, velocity and position at time , while and are, respectively, the initial velocity and initial position. ANSWER:
Express your answer in meters to three significant figures. ANSWER: =
Any time that you are working a physics problem, you should check that your answer is reasonable. Even when you don't have an applet with which to check, you have a wealth of personal experience. For example, if you obtain an answer such as "the distance from New York to Los Angeles is 3.96 ," you know it must be wrong. You should always try to relate situations from physics class to real-life situations.
Kinematic Vocabulary
Description: A series of questions designed to sharpen the understanding of terms used to describe motion. One of the difficulties in studying mechanics is that many common words are used with highly specific technical meanings, among them velocity, acceleratio n, position, speed, and displacement. The series of questions in this problem is designed to get you to try to think of these quantities like a physicist. Answer the questions in this problem using words from the following list: A. B. C. D. E. position direction displacement coordinates velocity
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F. G. H. I. J. K.
acceleration distance magnitude vector scalar components
Part A Velocity differs from speed in that velocity indicates a particle's __________ of motion. Enter the letter from the list given in the problem introduction that best completes the sentence. direction
Part B Unlike speed, velocity is a __________ quantity. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: I vector
Part C A vector has, by definition, both __________ and direction. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: H magnitude
Part D Once you have selected a coordinate system, you can express a two-dimensional vector using a pair of quantities known collectively as __________. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: K components D coordinates
Part E Speed differs from velocity in the same way that __________ differs from displacement. Hint E.1 Definition of displacement
Displacement is the vector that indicates the difference of two positions (e.g., the final position from the initial position). Being a vector, it is independent of the coordinate system used to describe it (although its vector components depend on the coordinate system). Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: G
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distance
Part F Consider a physical situation in which a particle moves from point A to point B. This process is described from two coordinate systems that are identical except that they have different origins. The __________ of the particle at point A differ(s) as expressed in one coordinate system compared to the other, but the __________ from A to B is/are the same as expressed in both coordinate systems. Type the letters from the list given in the problem introduction that best complete the sentence. Separate the letters with commas. There is more than one correct answer, but you should only enter one pair of comma-separated letters. For example, if the words "vector" and "scalar" fit best in the blanks, enter I,J. C position C D C coordinates C A displacement position displacement D displacement coordinates displacement A G position G D G coordinates G A distance position distance D distance coordinates distance A E position E D E
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coordinates E A velocity position velocity D velocity coordinates velocity A B position B D B coordinates B A direction position direction D direction coordinates direction
The coordinates of a point will depend on the coordinate system that is chosen, but there are several other quantities that are independent of the choice of origin for a coordinate system: in particular, distance, displacement, direction, and velocity. In working physics problems, unless you are interested in the position of an object or event relative to a specific origin, you can usually choose the coordinate system origin to be wherever is most convenient or intuitive. Note that the vector indicating a displacement from A to B is usually represented as .
Part G Identify the following physical quantities as scalars or vectors. ANSWER:
View
A Man Running to Catch a Bus
Description: This problem looks at the minimum speed to catch an accelerating bus with head start - uses discriminant. A man is running at speed (much less than the speed of light) to
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catch a bus already at a stop. At
, when he is a distance
from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus.
Part A What is Hint A.1 , the position of the man as a function of time? Which equation should you use for the man's speed? .
Because the man's speed is constant, you may use Answer symbolically in terms of the variables , , and . ANSWER:
=
Part B What is Hint B.1 , the position of the bus as a function of time? Which equation should you use for the bus's acceleration? .
Because the bus has constant acceleration, you may use Recall that . and .
Answer symbolically in terms of ANSWER: =
Part C What condition is necessary for the man to catch the bus? Assume he catches it at time Hint C.1 How to approach this problem , the man must arrive at the position of the door of the bus. .
If the man is to catch the bus, then at some moment in time How would you express this condition mathematically? ANSWER:
Part D Inserting the formulas you found for following: and into the condition , you obtain the
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, or
.
Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man's speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus.
Hint D.1
Consider the discriminant . What is the discriminant (the part under the radical) of the
Use the quadratic equation to solve: solution for ?
Hint D.1.1 The quadratic formula Recall: If then
ANSWER: =
Hint D.2 What is the constraint? To get a real value for exceed . and . , the discriminant must be greater then or equal to zero. This condition yields a constraint that
Express the minimum value for the man's speed in terms of ANSWER: =
Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint E.1 What is the general quadratic equation? , where , , and are constants. Depending on the value of the
The general quadratic equation is discriminant, , the equation may have , , or .
1. two real valued solutions if 2. one real valued solution if 3. two complex valued solutions if
In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: No; there is no chance he is going to get aboard. Yes; he will get a second chance
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A Flea in Flight
Description: The initial speed and time of flight for a flea's leap are calculated. In this problem, you will apply kinematic equations to a jumping flea. Take the magnitude of free-fall acceleration to be 9.80 . Ignore air resistance. Part A A flea jumps straight up to a maximum height of 0.480 Hint A.1 Finding the knowns and unknowns , and denote the . What is its initial velocity as it leaves the ground?
Take the positive y direction to be upward, the y coordinate of the initial position of the flea to be final height of the flea by , whose value 0.480 you know. Let height, its initial velocity, its final velocity (at maximum height), and following quantities is/are known? Hint A.1.1 The number of known quantities
be the duration of the flea's leap to its maximum its (constant) acceleration. Which of the
Typically, you need to know the values of four variables in order to solve any of the kinematic equations, because they contain five variables each, with the exception of , which contains only four variables, in which case you would need to know the values of only three of these variables. Since we may place the flea at any point of the y axis to begin its jump, we have conveniently assumed that is equal to 0. Hint A.1.2 What is the flea's velocity at its maximum height? What is the velocity of the flea at its maximum height of 0.480 ?
Express your answer in meters per second to three significant figures. ANSWER: =
Check all that apply. ANSWER:
Hint A.2
Determine which kinematic equation to use
Decide which kinematic equation makes the solution of this problem easiest. That is, look for an equation that contains the variable you are solving for and in which all the other variables are known. ANSWER:
Now substitute the known quantities into this equation and find
, the variable that you are looking for.
Hint A.3 Some algebra help You have determined that the simplest equation to use is . To solve for , you must first subtract the term from both sides of the equation, and then take the square root
of both sides. Keep in mind that the acceleration is negative.
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Express your answer in meters per second to three significant figures. ANSWER: =
Part B How long is the flea in the air from the time it jumps to the time it hits the ground? Hint B.1 How to approach the problem
One approach is to find the time it takes for the flea to go from the ground to its maximum height, and then find the time it takes for the flea to fall from its maximum height to the ground. The subsequent hints will guide you through this approach. Hint B.2 Find the time from the ground to the flea's maximum height What is the time )? Express your answer in seconds to three significant figures. ANSWER: = it takes the flea to go from the ground ( , ) to its maximum height ( 0.480 ,
Hint B.3 Find the time from the flea's maximum height to the ground What is the time )? Express your answer in seconds to three significant figures. ANSWER: = that it takes for the flea to fall from its maximum height ( 0.480 , ) to the ground (
Express your answer in seconds to three significant figures. ANSWER: time in air =
Notice that the time for the flea to rise to its maximum height is equal to the time it takes for it to fall from that height back to the ground. This is a general feature of projectile motion (any motion with ) when air resistance is neglected and the landing point is at the same height as the launch point. There is also a way to find the total time in the air in one step: just use
and realize that you are looking for the value of
for which
.
Clear the Runway
Description: Find how long it takes a plane to take off given the length of the runway and the constant acceleration of the plane. Also, find the velocity of the plane as it takes off, and the distance traveled in the first and last second of the plane's run. To take off from the ground, an airplane must reach a sufficiently high speed. The velocity required for the takeoff, the takeoff velocity, depends on several factors, including the weight of the aircraft and the wind velocity.
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Part A A plane accelerates from rest at a constant rate of 5.00 along a runway that is 1800 long. Assume that the plane
reaches the required takeoff velocity at the end of the runway. What is the time Hint A.1 How to approach the problem
needed to take off?
As the plane travels along the runway, it has constant acceleration. To solve the problem, you'll need to use the kinematics equations for such motion. In particular, you need to use the equation relating the distance traveled and time. Hint A.2 Find the equation for the distance traveled by the plane traveled by the plane during a certain interval of time ? Let and be,
Which expression best describes the distance accelerates from rest. ANSWER:
respectively, the initial position and speed of the plane, and use
for the acceleration of the plane. Remember that the plane
=
Express your answer in seconds. ANSWER: =
Part B What is the speed Hint B.1 of the plane as it takes off?
How to approach the problem
Since you are given the constant acceleration of the plane, and you have also found the time it takes to take off, you can calculate the speed of the plane as it ascends into the air using the equation for the velocity of an object in motion at constant acceleration. Hint B.2 Find the equation for the velocity of the plane of the plane after a certain interval of time ? Let be the initial velocity of for the acceleration of the plane. Remember that the plane starts from rest.
Which expression best describes the velocity the plane, and use ANSWER:
=
Alternatively, you can use the relation
(recalling that in this case
).
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Express your answer numerically at meters per second. ANSWER: =
Part C What is the distance Hint C.1 traveled by the plane in the first second of its run?
How to approach the problem
Apply the same equation that you used to solve Part A. Express your answer numerically in meters. ANSWER: =
Part D What is the distance Hint D.1 traveled by the plane in the last second before taking off?
How to approach the problem
Use the equation that gives the distance traveled as a function of time. Note that you are looking for the distance traveled in the last second before the plane takes off, which can be expressed as the length of the runway minus the distance traveled by the plane up to that last second. Express your answer numerically in meters. ANSWER: =
Since the plane is accelerating, the average speed of the plane during the last second of its run is greater than its average speed during the first second of the run. Not surprisingly, so is the distance traveled. Part E What percentage of the takeoff velocity did the plane gain when it reached the midpoint of the runway? Hint E.1 How to approach the problem
You need to find the velocity of the plane by the time it covers half the length of the runway and compare it with the takeoff velocity. Apply the same method that you used to determine the takeoff velocity. Express your answer numerically. ANSWER:
This is a "rule of thumb" generally used by pilots. Since the takeoff velocity for a particular aircraft can be computed before the flight, a pilot can determine whether the plane will successfully take off before the end of the runway by verifying that the plane has gained 70% of the takeoff velocity by the time it reaches half the length of the runway. If the plane hasn't reached that velocity, the pilot knows that there isn't enough time to reach the needed takeoff velocity before the plane reaches the end of the runaway. At that point, applying the brakes and aborting the takeoff is the safest course of action.
The Graph of a Sports Car's Velocity
Description: Find an object's acceleration and distance traveled from a graph of velocity as a function of time. The graph in the figure
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shows the velocity
of a sports car as a function of time . Use the graph to
answer the following questions.
Part A Find the maximum velocity Hint A.1 of the car during the ten-second interval depicted in the graph.
How to approach the problem
Because the graph displays the car's velocity at each moment in time, the maximum velocity of the car can be found simply by locating the maximum value of the velocity on the graph. Express your answer in meters per second to the nearest integer. ANSWER: =
Part B During which time interval is the acceleration positive? Hint B.1 Finding acceleration from the graph
Recall that acceleration is the rate of change of velocity with respect to time. Therefore, on this graph of velocity vs. time, acceleration is the slope of the graph. Recall that the slope is defined by for a graph of vs. , or in this case. If the graph is increasing from left to right, then the slope is positive. Indicate the best answer. ANSWER: to to to to to
Part C Find the maximum acceleration Hint C.1 of the car.
How to approach the problem with respect to time . In this problem, the car's velocity is vs. curve at that moment. If
The car's acceleration is the rate of change of the car's velocity the vs.
given graphically, so the car's acceleration at a given moment is found from the slope of the
curve over some time interval is represented by a straight line, the instantaneous acceleration anywhere in that
interval is equal to the slope of the line, that is, to the average acceleration over that time interval. To find the maximum acceleration, find the value of the curve's greatest positive slope. Hint C.2 Find the final velocity on the interval with greatest acceleration The slope of the curve is greatest during the first second of motion. The slope of the graph on this interval is given by the change in velocity divided by the change in time over the interval from to . At time , the car's velocity is zero. Find the velocity of the car at time .
Express your answer in meters per second to the nearest integer. ANSWER:
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ANSWER:
=
Express your answer in meters per second per second to the nearest integer. ANSWER: =
Part D Find the minimum magnitude of the acceleration Hint D.1 How to approach the problem of the car.
To find the minimum magnitude of the acceleration of the car, you must find the point where the absolute value of the slope is smallest. Express your answer in meters per second per second to the nearest integer. ANSWER: =
Part E Find the distance Hint E.1 traveled by the car between and .
How to approach the problem
In this problem, the car's velocity as a function of time is given graphically, so the distance traveled is represented by the area under the vs. graph between and . Hint E.2 Find the distance traveled in the first second What is the distance Hint E.2.1 traveled between The area of a triangle and ?
Observe that the region in question is a triangle , whose area is therefore one-half the product of the base and the height.
Express your answer in meters. ANSWER: d_0,1 =
Hint E.3 Find the distance traveled in the second second What is the distance Hint E.3.1 traveled between The shape of the region and ?
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The region under the graph between 1 and 2 seconds can be seen as consisting of a rectangle and a triangle.
Express your answer in meters. ANSWER: =
Express your answer in meters to the nearest integer. ANSWER: =
Score Summary:
Your score on this assignment is 0%. You received 0 out of a possible total of 120 points.
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