chapter 2 skill building

MasteringPhysics: Assignment Print View

Given Positions, Find 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) 0 1 2 3 4 5 6 7 8 9

x (m) 0 1 4 9 16 24 32 40 46 48 Part A

Which graph best represents the function x(t), describing the object's position vs. time? ANSWER: 1 2 3 4

Part B

http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1040686 (1 of 26)4/11/2006 2:36:59 PM


Which of the following graphs best represents the function v(t), describing the object's velocity as a function of time? Part B.1

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Hint B.2

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Part B.3

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Part B.4

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ANSWER: 1 2 3 4

Part C

Which of the following graphs best represents the function a(t), describing the acceleration of this object? Part C.1

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Part C.2

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Hint C.3

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ANSWER: 1 2 3 4

A PSS is a problem solving strategy. They appear throughout the text. Pay serious attention to learning how to apply the strategy to all of your homework problems. In particular, get in the habit of using motion diagrams, and pictorial and graphical representations.

PSS 2.1: Can't Hit the Can!



Learning Goal: To practice Problem-Solving Strategy 2.1 for problems involving kinematics with constant acceleration.A car is traveling at a constant velocity of magnitude when the driver notices a garbage can on the road in front of him. At that moment, the distance between the garbage can and the front of the car is . A time after noticing the garbage can, the driver applies the brakes and slows down at a

constant rate before coming to a halt just before the garbage can. What is the magnitude of the car's acceleration after the brakes are applied?

MODEL: Use the particle model. Make other appropriate simplifying assumptions.

VISUALIZE: Use different representations of the information in the problem.

■ Draw a motion diagram. Motion diagrams are part of the physical representation. ■ Draw a pictorial representation. This helps you assess the information you are given and

starts the process of translating that information into mathematical symbols. ■ Use a graphical representation if appropriate for the problem. ■ Go back and forth between these three representations as needed.

SOLVE: The mathematical representation is based on the three kinematic equations

,

,

and

.

■ Use or , as appropriate to the problem, rather than the generic . ■ If necessary, replace subscripts i and f with numerical subscripts defined in the pictorial

representation. ■ Recall that, for uniform motion with constant velocity, .

ASSESS: Is your result believable? Does it have appropriate units? Does it make sense?

Start by making simplifying assumptions appropriate for the situation.

Part A



The car should be treated as

ANSWER: a solid macroscopic object of certain shape and size.

a pointlike particle.

a thin straight line.

Now draw a motion diagram and pictorial representation, including all the elements listed in the problem-solving strategy. Use your sketch to answer the following questions.

Part B

Consider the incomplete pictorial representation shown here: . Missing from this figure are the arrows representing the acceleration of the car for the two different segments of the motion. These arrows should appear below the symbols and .

For each segment of the motion, indicate whether the acceleration of the car should be drawn as pointing to the right, as pointing to the left, or as having zero magnitude (and therefore no direction).

Enter R for right, L for left, or 0 for each of the two accelerations. Separate your two answers with a comma.

ANSWER: 0,L

The list of known and unknown quantities are also missing from the diagram. You'll need to figure these out for yourself before you go on.

Part C



Which diagram correctly shows position, velocity, and acceleration of the moving car after the brakes are applied?

ANSWER: a b c d

Part D

Diagram (a) in the previous part is incorrect because it leads to which of the following incorrect conclusions about the motion of the car?

A. The car is moving at constant speed. B. The car is speeding up. C. The car is not moving in a straight line. D. The direction of acceleration of the car does not agree with the velocity vectors. E. The acceleration of the car is decreasing. F. The acceleration of the car is increasing. G. The acceleration of the car is changing direction.

List alphabetically the letters corresponding to the statements describing the incorrect implications of diagram (a). Do not use commas. For instance, if you think that statements A, B, and C describe the errors in the diagram, enter ABC.

ANSWER: BD

Part E



Of the following list of variables, which represent known quantities?

A. B. C.

D. E. F.

Enter the letters of the correct answers in alphabetical order. Do not use commas.

ANSWER: BCDE

Among the other variables in the pictorial representation provided in Part B, is a known quantity and you can assume that and .

Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution.

Part F

Find , the magnitude of the acceleration of the car.

Part F.1

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Hint F.2

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Express the magnitude of the acceleration in terms of the variables given in the problem introduction: , , and . You may or may not use all of them.

ANSWER: =

When you work on a problem on your own, without the computer-provided feedback, only you can assess whether your answer seems right. The following questions will help you practice the skills necessary for such an assessment.

Part G



Which of the following algebraic expressions have the dimensions of acceleration?

A.

B.

C.

D.

E.

List alphabetically the letters corresponding to all the expressions that have the correct dimensions. Do not use commas. For instance, if you think that expressions A, B, and C have the correct dimensions, enter ABC.

ANSWER: ABDE

Part H

Imagine a situation exactly as given in the problem introduction except that the driver continues for a longer time between noticing the can and hitting the brakes. If he is still to stop just before hitting the can, how would the magnitude of his acceleration compare to that derived in the Solve step above?

ANSWER: It would be the same. It would be greater. It would be smaller.

From the answer you derived, you can see that if increases, then the magnitude of the acceleration will increase. Of course, the driver cannot wait so long before hitting the brakes that

; otherwise he will already have collided with the can.

Part I

Imagine a situation exactly as given in the problem introduction except that the driver notices the garbage can from farther away. If he is still to stop just before hitting the can, how would the magnitude of his acceleration compare to that derived in the Solve step above?

ANSWER: It would be the same. It would be greater. It would be smaller.

From the answer you derived, you can see that if increases, then the magnitude of the

acceleration will decrease.



What Velocity vs. Time Graphs Can Tell You

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

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Hint A.2

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Express your answer in meters per second.

ANSWER: = 0.500

Part B

What is the total distance traveled by the particle?

Hint B.1

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Part B.2

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Part B.3

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Part B.4

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Express your answer in meters.



ANSWER: = 75.0

Part C

What is the average acceleration of the particle over the first 20.0 seconds?

Hint C.1

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Hint C.2

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Express your answer in meters per second per second.

ANSWER: =

7.50×10−2

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 of the particle at ?

Hint D.1

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Hint D.2

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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

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Part E.2

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Part E.3

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Part E.4

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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.

Kinematic Vocabulary

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. position B. direction C. displacement D. coordinates E. velocity F. acceleration G. distance H. magnitude I. vector J. scalar

K. 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.

ANSWER: B

Part B

Unlike speed, velocity is a __________ quantity.


ANSWER: I

Part C



A vector has, by definition, both __________ and direction.


ANSWER: H

Part D

Once you have selected a coordinate system, you can express a two-dimensional vector using a pair of quantities known collectively as __________.


ANSWER: D

Part E

Speed differs from velocity in the same way that __________ differs from displacement.

Hint E.1

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ANSWER: G

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.

ANSWER: A,C

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

Which of the following physical quantities are scalars?

A. position B. velocity C. displacement D. speed E. acceleration F. average velocity G. distance

Type the letters of all the correct answers in alphabetical order. For this question do not separate your answers with commas.

ANSWER: DG

The other quantities are vectors.

One-Dimensional Kinematics with Constant 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;

● 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 is a function of time (i.e., is not constant).

ANSWER: true

false

Part B


ANSWER: true

false

Recall that represents an initial value, not a variable. It refers to the position of an object at some initial moment.

Part C


ANSWER: true

false

Part D


ANSWER: true

false

The velocity always varies with time when the linear acceleration is nonzero.

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 represents the particle's velocity at what instant in time?

ANSWER: at time

at the "initial" time

when a time has passed since the particle's velocity was

More generally, the equations of motion can be written as

and

.

Here is the time that has elapsed since the beginning of the particle's motion, that is,

, where is the current time and is the time at which we start measuring the particle's

motion. The terms and 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 . At time , particle B has

twice the acceleration, half the velocity, and the same position that particle A had at time .



Part G

What is the equation describing the position of particle B?

Hint G.1

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ANSWER:

Part H

At what time does the velocity of particle B equal that of particle A?

Part H.1

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Part H.2

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ANSWER:

The two particles never have the same velocity.



What x vs. t Graphs Can Tell You

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. Refer to this graph to answer Parts A, B, and C.

Part A

What is the total distance traveled by

the particle?

Hint A.1

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Hint A.2

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ANSWER: = 30.0

Part B

What is the average velocity of the particle over the time interval ?

Hint B.1

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Hint B.2

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ANSWER: = 0.600

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.

Part C



What is the instantaneous velocity of the particle at ?

Hint C.1

Hint not displayed


ANSWER: = 0.600

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

Hint not displayed



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.

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

Hint not displayed


ANSWER: = 30.0

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 .

A Flower Pot Falling Past a Window

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 above the bottom of your window was the flower pot dropped?

Hint A.1

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Part A.2

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Hint A.3

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Express your answer in terms of , , and .



ANSWER: =

Part B

If the bottom of your window is a height above the ground, what is the velocity of the

pot as it hits the ground? You may introduce the new variable , the speed at the bottom of the window, defined by

.

Hint B.1

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Part B.2

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Express your answer in terms of some or all of the variables , , , , and .

ANSWER: =

Tossing Balls off a Cliff

Learning Goal: To clarify the distinction between speed and velocity, and to review qualitatively one-dimensional kinematics.A woman stands at the edge of a cliff, holding one ball in each hand. At time , she throws one ball straight up with speed and the other straight down, also with speed .

For the following questions neglect air resistance. Pay particular attention to whether the answer involves "absolute" quantities that have only magnitude (e.g., speed) or quantities that can have either sign (e.g., velocity). Take upward to be the positive direction.

Part A



If the ball that is thrown downward has an acceleration of magnitude at the instant of its release (i.e., when there is no longer any force on the ball due to the woman's hand), what is the relationship between and , the magnitude of the acceleration of gravity?

ANSWER:

Part B

Which ball has the greater acceleration at the instant of release?

ANSWER: the ball thrown upward

the ball thrown downward

Neither; the accelerations of both balls are the same.

Part C

Which ball has the greater speed at the instant of release?

Hint C.1

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Neither; the speeds are the same.

Part D

Which ball has the greater average speed during the 1-s interval after release (assuming neither hits the ground during that time)?

Hint D.1

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Neither; the average speeds of both balls are the same.



Part E

Which ball hits the ground with greater speed?



Neither; the balls hit the ground with the same speed.

Analyzing Position versus Time Graphs: Conceptual Question

Two cars travel on the parallel lanes of a two-lane road. The cars’ motions are represented by the position versus time graph shown in the figure. Answer the questions using the times from the graph indicated by letters.

Part A

At which of the times do the two cars pass?

Hint A.1

Hint not displayed

ANSWER:

Part B

Are the two cars traveling in the same direction when they pass?

ANSWER: yes no

Part C



At which of the lettered times, if any, does car #1 momentarily stop?

Hint C.1

Hint not displayed

ANSWER:

Part D

At which of the lettered times, if any, does car #2 momentarily stop?

Hint D.1

Hint not displayed

ANSWER:

Part E

At which of the lettered times are the cars moving at approximately the same velocity?

Hint E.1

Hint not displayed

ANSWER:

Velocity from Graphs of Position versus Time

An object moves along the x axis during four separate trials. Graphs of position versus time for each trial are shown in the figure.



Part A

During which trial or trials is the object's velocity not constant?

A. Trial A B. Trial B C. Trial C D. Trial D

Hint A.1

Hint not displayed

Hint A.2

Hint not displayed

Enter the letters of the correct statement(s) in alphabetical order. For example, if A and C are correct, enter AC.

ANSWER: B

The graph of the motion during Trail B has a changing slope and therefore is not constant. The other trials all have graphs with constant slope and thus correspond to motion with constant velocity.

Part B

During which trial or trials does the object have the greatest average velocity?

A. Trial A B. Trial B C. Trial C D. Trial D

Hint B.1

Hint not displayed

Enter the letters of the correct statement(s) in alphabetical order. For example, if A and C are correct, enter AC.

ANSWER: B



You recognized that although the magnitudes of the average velocity in Trial B and Trial D are equal, their directions are opposite. This makes the average velocity in Trial D less than the average velocity in Trial B. The object does not move during Trial C, so it has an average velocity of zero. During Trial A the object has a positive average velocity but its magnitude is less than that in Trial B.


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