Impulse on a Baseball

Learning Goal: To understand the relationship between force, impulse, and momentum.

The effect of a net force acting on an object is related both to the force and to the total time the

force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by

.

The impulse is a vector pointing in the same direction as the force vector. The units of are or

.

Recall that when a net force acts on an object, the object will accelerate, causing a change in its

velocity. Hence the object's momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object's motion:

.

So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time.

In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a

good solid hit. His swing applies a force of 12,000 to the ball for a time of .

Part A

Assuming that this force is constant, what is the magnitude of the impulse on the ball?

Enter your answer numerically in newton seconds.

ANSWER: =

8.40Correct

We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure.

Part B

The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning.

ANSWER: For this graph, the

Top of FormCorrect

Bottom of Formof the rectangle corresponds to the impulse.

The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant.

A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph

area

in the figure.

Part C

If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common?

ANSWER:

maximum force

area

slope

Correct

When the net force varies over time, as in the case of the real net force acting on the baseball, you

can simplify the problem by finding the average net force acting on the baseball during time

. This average net force is treated as a constant force that acts on the ball for time . The

impulse on the ball can then be found as .

Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively.

The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known,

the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem.

Part D

Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction.

ANSWER: The impulse on the ball caused by the bat will be in the

Top of FormCorrect

Bottom of Formx direction.

Part E

Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a

speed of 32 in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's velocity just after leaving the bat if the bat

negative

applies an impulse of to the baseball?

Enter your answer numerically in meters per second.

ANSWER: =

-25.9Correct

The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive.

A Ball Hits a Wall Elastically

A ball of mass moving with velocity strikes a vertical wall.

The angle between the ball's initial velocity

vector and the wall is as shown on the diagram, which depicts the situation as seen from above.

The duration of the collision between the ball and the wall is , and this collision is completely elastic. Friction is negligible, so the ball does not start spinning. In this idealized collision, the force exerted on the ball by the wall is parallel to the x axis.

Part A

What is the final angle that the ball's velocity vector makes with the negative y axis?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the y component of the ball's final velocity

Hint not displayed

Hint A.3 Find the component of the ball's final velocity

Hint not displayed

Hint A.4 Putting it together

Hint not displayed

Express your answer in terms of quantities given in the problem introduction.

ANSWER:

= Correct

Part B

What is the magnitude of the average force exerted on the ball by the wall?

Hint B.1 What physical principle to use

Hint not displayed

Hint B.2 Change in momentum of the ball

Hint not displayed

Express your answer in terms of variables given in the problem introduction and/or .

ANSWER: =

Correct

A One-Dimensional Inelastic Collision

Block 1, of mass = 4.10 , moves along a frictionless air track with speed = 13.0 . It

collides with block 2, of mass = 53.0 , which was initially at rest. The blocks stick together

after the collision.

Part A

Find the magnitude of the total initial momentum of the two-block system.

Hint A.1 How to approach the problem

Hint not displayed

Express your answer numerically.

ANSWER: =

53.3Correct

Part B

Find , the magnitude of the final velocity of the two-block system.

Hint B.1 How to approach the problem

Hint not displayed

Express your answer numerically.

ANSWER: =

0.933Correct

Part C

What is the change in the system's kinetic energy due to the collision?

Hint C.1 Find the initial kinetic energy

Hint not displayed

Express your answer numerically in joules.

ANSWER: =

-322Correct

J

Collisions in One Dimension

On a frictionless horizontal air table, puck A (with mass 0.252 ) is moving toward puck B (with

mass 0.370 ), which is initially at rest. After the collision, puck A has velocity 0.118 to the

left, and puck B has velocity 0.648 to the right.

Part A

What was the speed of puck A before the collision?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 The initial momentum

Hint not displayed

Hint A.3 Find the final momentum of puck A

Hint not displayed

Hint A.4 Find the final momentum of puck B

Hint not displayed

ANSWER: =

0.833Correct

Part B

Calculate , the change in the total kinetic energy of the system that occurs during the collision.

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Find the initial kinetic energy of puck A

Hint not displayed

Hint B.3 Find the final kinetic energy of puck A

Hint not displayed

Hint B.4 Find the final kinetic energy of puck B

Hint not displayed

ANSWER:

=

−8.08×10−3Correct

A Bullet Is Fired into a Wooden Block

A bullet of mass is fired horizontally with speed at a wooden block of mass resting on a frictionless table. The bullet hits the block and becomes completely embedded within it. After the bullet has come to rest within the block, the block, with the bullet in it, is traveling at speed .

Part A

Which of the following best describes this collision?

Hint A.1 Types of collisions

Hint not displayed

ANSWER: perfectly elastic

partially inelastic

perfectly inelastic

Correct

Part B

Which of the following quantities, if any, are conserved during this collision?

Hint B.1 When is kinetic energy conserved?

Hint not displayed

ANSWER: kinetic energy only

momentum only

kinetic energy and momentum

neither momentum nor kinetic energy

Correct

Part C

What is the speed of the block/bullet system after the collision?

Hint C.1 Find the momentum after the collision

Hint not displayed

Hint C.2 Use conservation of momentum

Hint not displayed

Express your answer in terms of , , and .

ANSWER: =

Correct

Problem 9.13

A child in a boat throws a 5.35- package out horizontally with a speed of 10.0 .

Part A

Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of

the child is 23.0 and that of the boat is 35.0 . (Take the package's direction of motion as positive.)

ANSWER:

-0.922

= Correct

Problem 9.35

A 0.475- hockey puck, moving east with a speed of 5.30 , has a head-on collision with a

0.880- puck initially at rest.

Part A

Assuming a perfectly elastic collision, what will be the speed of each object after the collision?

Enter your answers numerically separated by a comma.

ANSWER: , =

1.58,3.72All attempts used; correct answer displayed

Part B

What will be the direction of the lighter object after the collision.

ANSWER:

West

East

Correct

Part C

What will be the direction of the heavier object after the collision.

ANSWER:

West

East

Correct

Problem 9.51

A bullet of mass 1.7×10−3 embeds itself in a wooden block with mass 0.999 , which then

compresses a spring ( = 160 ) by a distance 5.0×10−2 before coming to rest. The coefficient of kinetic friction between the block and table is 0.60.

Part A

What is the initial speed of the bullet?

Express your answer using two significant figures.

ANSWER: =

590Correct

Part B

What fraction of the bullet's initial kinetic energy is dissipated (in damage to the wooden block, rising temperature, etc.) in the collision between the bullet and the block?

Express your answer using two significant figures.

ANSWER: =

1.0Correct

Problem 9.89

A gun fires a bullet vertically into a 1.40- block of wood at rest on a thin horizontal sheet.

Part A

If the bullet has a mass of 18.5 and a speed of 310 , how high will the block rise into the air after the bullet becomes embedded in it?

ANSWER:

=

0.834Correct

Problem 9.99

Two balls, of masses = 46 and = 64 , are suspended as shown in the figure. The lighter

ball is pulled away to a 66 angle with the vertical and released. Assume that the positive axis is

directed to the right.

Part A

What is the velocity of the lighter ball before impact?

Express your answer using two significant figures.

ANSWER: =

1.9All attempts used; correct answer displayed

Part B

What is the velocity of each ball after the elastic collision?

Express your answers using two significant figures. Enter your answers numerically separated by a comma.

ANSWER: ,

=

-0.31,1.6Correct

Part C

What will be the maximum height of each ball after the elastic collision?

Express your answers using two significant figures. Enter your answers numerically separated by a comma.

ANSWER: , =

4.9×10−3,0.13Correct

Problem 9.100

A block of mass 2.50 slides down a 30.0 incline which is 3.60 high. At the bottom, it strikes a

block of mass 6.05 which is at rest on a horizontal surface. (Assume a smooth transition at the

bottom of the incline.)

Part A

If the collision is elastic, and friction can be ignored, determine the speed of the block with mass

2.50 after the collision.

ANSWER: =

3.49Correct

Part B

If the collision is elastic, and friction can be ignored, determine the speed of the blockwith mass

6.05 after the collision.

ANSWER:

=

4.91Correct

Part C

How far back up the incline the smaller mass will go.

ANSWER:

=

1.24Correct

Collision at an Angle

Two cars, both of mass , collide and stick together. Prior to the collision, one car had been

traveling north at speed , while the second was traveling at speed at an angle south of east (as

indicated in the figure). After the collision, the two-car system travels at speed at an angle

east of north.

Part A

Find the speed of the joined cars after the collision.

Hint A.1 Determine the conserved quantities

Hint not displayed

Hint A.2 The component of the final velocity in the east-west direction

Hint not displayed

Hint A.3 Find the north-south component of the final momentum

Hint not displayed

Hint A.4 Math help

Hint not displayed

Express your answer in terms of and .

ANSWER: =

Correct

Part B

What is the angle with respect to north made by the velocity vector of the two cars after the collision?

Hint B.1 A formula for

Hint not displayed

Express your answer in terms of . Your answer should contain an inverse trigonometric function.

ANSWER:

=All attempts used; correct answer displayed