Mastering Physics Hw 6
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Transcript of Mastering Physics Hw 6
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