# Physics

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 12/29

time?

**Hint 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:

**Hint 2. **What are the implications of zero velocity?

Two of the possible velocity vs. time graphs indicate zero velocity between and . What would

the corresponding position vs. time graph look like in this region?

ANSWER:

**Hint 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:

positive and increasing

positive and decreasing

negative and increasing

negative and decreasing

0 – 0 – T

a horizontal line

straight but sloping up to the right

straight but sloping down to the right

curved upward

curved downward

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 13/29

ANSWER:

Correct

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

Which of the following graphs best represents the function

, describing the acceleration of this object?

has spikes

has no discontinuities

has no abrupt changes of slope

is constant

1

2

3

4

2BWH<00>-

40Ã40

0Ã0

2BWH <0 0 >

0 – 0

0

0

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 14/29

**Hint 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:

**Hint 2. **Calculate the acceleration in the region of constant velocity

What is the acceleration over the interval during which the object travels at constant speed?

Answer numerically in meters per second squared.

ANSWER:

**Hint 3. **Find the initial acceleration

Acceleration is the time derivative of velocity. Initially the acceleration is _________.

ANSWER:

ANSWER:

zero

positive

negative

= 0 NT

zero

positive

negative

1

2

3

4

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 15/29

Correct

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 and within the

interval to compute the average.

Velocity and Acceleration of a Power Ball

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.

**Hint 1. **Velocity and displacement vectors

BWH<00>-

20Ã20

0Ã0

0 0

. . 0

2 . 0 . 0

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 16/29

By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the

vector displacement as the distance traveled by the ball, the length of is directly proportional to the

length of . Since the length of displacement vectors is increasing, so is the length of velocity vectors.

ANSWER:

Correct

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 displacement vectors, as shown in the figure here . Indicate whether the

velocity and acceleration of the ball are, respectively,

positive (upward), negative, or zero.

Use P, N, and Z for positive (upward), negative, and

zero, respectively. Separate the letters for velocity

and acceleration with a comma.

**Hint 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 .

ANSWER:

Correct

.. 2.

..

increasing

decreasing

not changing

2. ..

2.- 2 . 0 Ã2 . 0

0 – 0 Ã0

N,N

11/10/2014 HW_Week2

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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.

**Hint 1. **Velocity and displacement vectors

By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the

vector displacement as the distance traveled by the ball, the length of is directly proportional to the

length of . Since the length of displacement vectors is decreasing, so is the length of velocity vectors.

ANSWER:

Correct

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.

Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and

acceleration with a comma.

.. 2.

..

increasing

decreasing

not changing

11/10/2014 HW_Week2

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 18/29

**Hint 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 .

ANSWER:

Correct

Part E