PC1431 MasteringPhysics Assignment 4

Assignment 4: Linear Momentum Due: 2:00am on Saturday, October 9, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View]

A Game of Frictionless Catch Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass

and throws it to Jackie, who catches it. Assume that the ball

travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is . Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie's speed relative to the ground after she catches the ball is . When answering the questions in this problem, keep the following in mind: of Chuck and his cart does not include the mass of the ball. The original mass The speed of an object is the magnitude of its velocity. An object's speed will always be a nonnegative quantity.

Part A Find the relative speed Hint A.1

between Chuck and the ball after Chuck has thrown the ball.

How to approach the problem Hint not displayed

Express the speed in terms of ANSWER:

=

and

.

Correct

Make sure you understand this result; the concept of "relative speed" is important. In general, if two objects are moving in opposite directions (either toward each other or away from each other), the relative speed between them is equal to the sum of their speeds with respect to the ground. If two objects are moving in the same direction, then the relative speed between them is the absolute value of the difference of the their two speeds with respect to the ground. Part B What is the speed Hint B.1

of the ball (relative to the ground) while it is in the air?

How to approach the problem Hint not displayed

Hint B.2

Initial momentum of Chuck, his cart, and the ball Hint not displayed

Hint B.3

Find the final momentum of Chuck, his cart, and the thrown ball Hint not displayed

Express your answer in terms of

,

, and

.

ANSWER: =

Correct

Part C What is Chuck's speed Hint C.1

(relative to the ground) after he throws the ball?

How to approach the problem Hint not displayed

Express your answer in terms of

,

, and

.

ANSWER: =

Correct

Part D Find Jackie's speed Hint D.1

(relative to the ground) after she catches the ball, in terms of

.

How to approach the problem Hint not displayed

Hint D.2

Initial momentum Hint not displayed

Hint D.3

Find the final momentum Hint not displayed

Express

in terms of

,

, and

.

ANSWER: =

Correct

Part E Find Jackie's speed Hint E.1

(relative to the ground) after she catches the ball, in terms of

How to approach the problem Hint not displayed

.

Express

in terms of

,

, and

.

ANSWER: =

Correct

A Girl on a Trampoline A girl of mass

kilograms springs from a trampoline with an initial upward velocity of

meters per second. At height

meters above the trampoline, the girl grabs a box of mass

kilograms. For this problem, use

meters per

second per second for the magnitude of the acceleration due to gravity.

Part A What is the speed Hint A.1

of the girl immediately before she grabs the box?

How to approach the problem Hint not displayed

Hint A.2

Initial kinetic energy Hint not displayed

Hint A.3

Potential energy at height Hint not displayed

Express your answer numerically in meters per second. ANSWER:

= 4.98 Correct

Part B What is the speed Hint B.1

of the girl immediately after she grabs the box?

How to approach the problem

Hint B.1

How to approach the problem Hint not displayed

Hint B.2

Total initial momentum Hint not displayed

Express your answer numerically in meters per second. ANSWER:

= 3.98 Correct

Part C Is this "collision" elastic or inelastic? Hint C.1

Definition of an inelastic collision Hint not displayed

ANSWER:

elastic inelastic Correct

In inelastic collisions, some of the system's kinetic energy is lost. In this case the kinetic energy lost is converted to heat energy in the girl's muscles as she grabs the box, and sound energy. Part D What is the maximum height

that the girl (with box) reaches? Measure

top of the trampoline. Hint D.1

How to approach the problem Hint not displayed

Hint D.2

Finding Hint not displayed

Hint D.3

Finding Hint not displayed

Express your answer numerically in meters. ANSWER:

= 2.81 Correct

Filling the Boat

with respect to the

A boat of mass 250

is coasting, with its engine in neutral, through the water at speed 3.00

when it starts to rain. The rain is falling vertically, and it accumulates in the boat at the rate of 10.0 . Part A What is the speed of the boat after time 2.00

has passed? Assume that the water resistance is

negligible. Hint A.1

How to approach the problem Hint not displayed

Hint A.2

Find the momentum of the boat before it starts to rain Hint not displayed

Hint A.3

Find the mass of the boat after it has started to rain Hint not displayed

Express your answer in meters per second. ANSWER:

2.78 Correct

Part B Now assume that the boat is subject to a drag force

due to water resistance. Is the component of

the total momentum of the system parallel to the direction of motion still conserved? ANSWER:

yes no Correct

The boat is subject to an external force, the drag force due to water resistance, and therefore its momentum is not conserved. Part C The drag is proportional to the square of the speed of the boat, in the form acceleration of the boat just after the rain starts? Take the positive

axis along the direction of

motion. Hint C.1

How to approach the problem Hint not displayed

Hint C.2

Find the time rate of change of momentum of the boat Hint not displayed

Express your answer in meters per second per second.

. What is the

Express your answer in meters per second per second. ANSWER:

−1.80×10−2 Correct

Rocket Car A rocket car is developed to break the land speed record along a salt flat in Utah. However, the safety of the driver must be considered, so the acceleration of the car must not exceed (or five times the acceleration of gravity) during the test. Using the latest materials and technology, the total mass of the car (including the fuel) is 6000 kilograms, and the mass of the fuel is one-third of the total mass of the car (i.e., 2000 killograms). The car is moved to the starting line (and left at rest), at which time the rocket is ignited. The rocket fuel is expelled at a constant speed of 900 meters per second relative to the car, and is burned at a constant rate until used up, which takes only 15 seconds. Ignore all effects of friction in this problem. Part A Find the acceleration Hint A.1

of the car just after the rocket is ignited.

How to approach the problem

The equation for the acceleration due to rocket propulsion is

, where

is the

exhaust speed. To use this equation, first find an expression for the rate of mass loss of the car. Hint A.2

find the rate of mass change

Find the rate

that the rocket car's mass is changing.

Express your answer to three significant figures. ANSWER:

= -133 Correct

Express your answer to two significant figures. ANSWER:

= 20 Correct

The driver of this car is experiencing just over

, or two times the acceleration one normally

feels due to gravity, at the start of the trip. This is not much different from the acceleration typically experienced by thrill seekers on a roller coaster, so the driver is in no danger on this score. Part B Find the final acceleration Hint B.1

of the car as the rocket is just about to use up its fuel supply.

What has changed? Hint not displayed

Find the final mass

Hint B.2

Hint not displayed Express your answer to two significant figures. ANSWER:

= 30 Correct

The driver of this car is experiencing just over

, or three times the acceleration one normally

feels due to gravity, by the end of the trip. This is the maximum acceleration achieved during the trip, and it is still very safe for the driver, who can easily withstand over with training.

Part C Find the final velocity

of the car just as the rocket is about to use up its fuel supply.

Find the change in speed

Hint C.1

Write an expression for the change in speed of the car from start to finish:

. You will need

to make use of the differential equation for rocket motion , if you don't know the equation for velocity of a rocket. Hint C.1.1 How to solve the differential equation

Hint not displayed Express your answer in terms of the exhaust speed fuel)

, and the final mass of the car

ANSWER:

, the initial mass of the car (plus

.

= Answer not displayed

Express your answer to two significant figures. ANSWER:

= 360 Correct

At the end of the trip, the driver is going a bit over Mach 1, or one times the speed of sound. This problem was based loosely on the breaking of the sound barrier by the ThrustSSC team in October 1997.

Three-Block Inelastic Collision A block of mass block of mass

moving with speed

undergoes a completely inelastic collision with a stationary

. The blocks then move, stuck together, at speed

system collides inelastically with a third block, of mass then move, stuck together, with speed

. All

three blocks have nonzero mass. Assume that the blocks slide without friction.

. After a short time, the two-block

, which is initially stationary. The three blocks

Part A Find

, the ratio of the velocity

the block of mass Hint A.1

of the two-block system after the first collision to the velocity

of

before the collision.

What physical principle to use Hint not displayed

Express your answer in terms of ANSWER:

,

, and/or

.

= Answer not displayed

Part B Find

, the ratio of the kinetic energy

kinetic energy Hint B.1

of the block of mass

of the two-block system after the first collision to the before the collision.

Formula for kinetic energy Hint not displayed

Express your answer in terms of ANSWER:

,

, and/or

.

= Answer not displayed

Part C Find

, the ratio of the velocity

of the block of mass Hint C.1

of the three-block system after the second collision to the velocity

before the collisions.

Total mass of the blocks Hint not displayed

Express your answer in terms of

,

, and/or

.

ANSWER:

= Answer not displayed

Part D Find

, the ratio of the kinetic energy

initial kinetic energy

of the block of mass

Express your answer in terms of ANSWER:

of the three-block system after the second collision to the

,

before the collisions.

, and/or

.

= Answer not displayed

Part E Suppose a fourth block, of mass speed

, is included in the series, so that the three-block system with

collides with the fourth, stationary, block. Find

, the ratio of the kinetic energy of the block of mass

the blocks after the final collision to the initial kinetic energy

of all

before any of

the collisions. Hint E.1

How to approach the question Hint not displayed

Express your answer in terms of ANSWER:

,

,

, and/or

.

= Answer not displayed

Conservation of Momentum in Two Dimensions Ranking Task Part A The figures below show bird's-eye views of six automobile crashes an instant before they occur. The automobiles have different masses and incoming velocities as shown. After impact, the automobiles remain joined together and skid to rest in the direction shown by . Rank these crashes according to the angle , measured counterclockwise as shown, at which the wreckage initially skids. Hint A.1

Conservation of momentum in two dimensions Hint not displayed

Hint A.2

Determining the angle Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.

Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:

Answer not displayed

Surprising Exploding Firework A mortar fires a shell of mass

at speed

. The shell explodes at the top of its trajectory (shown by a

star in the figure) as designed. However, rather than creating a shower of colored flares, it breaks into just two pieces, a smaller piece of mass

and a larger piece of mass

. Both pieces land at

exactly the same time. The smaller piece lands perilously close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance from the mortar. If there had been no explosion, the shell would have landed a distance

from the mortar. Assume that air resistance and the mass of the

shell's explosive charge are negligible.

Part A Find the distance Hint A.1

from the mortar at which the larger piece of the shell lands.

Find the position of the center of mass in terms of Hint not displayed

Hint A.2

Find the position of the center of mass in terms of Hint not displayed

Express ANSWER:

in terms of . = Answer not displayed

Pucks on Ice Two hockey players, Aaron and Brunnhilde, are pushing two pucks on a frictionless ice rink. The pucks are initially at rest on the starting line. Brunnhilde is pushing puck B, which has a mass three times as great as that of puck A, which Aaron is pushing. The players exert equal constant forces of magnitude on their pucks, directed horizontally, towards the finish line. They start pushing at the same time, and each player pushes his or her puck until it crosses the finish line, a distance away.

Part A Which puck reaches the finish line first? Hint A.1

Compute the relative acceleration of the pucks Hint not displayed

ANSWER:

Both pucks reach the finish line at the same time. Puck A reaches the finish line first. Puck B reaches the finish line first. More information is needed to answer this question.

Part B

Answer not displayed

Part B Let

be the magnitude of the kinetic energy of puck A at the instant it reaches the finish line.

Similarly,

is the magnitude of the kinetic energy of puck B at the (possibly different) instant it

reaches the finish line. Which of the following statements is true? Hint B.1

Determine the simplest way to answer this question Hint not displayed

Hint B.2

Work done on puck A Hint not displayed

Hint B.3

Work done on puck B Hint not displayed

ANSWER:

Answer not displayed

You need more information to decide.

Part C

Part not displayed

A Rocket in Deep Space A rocket is fired in deep space, where gravity is negligible. In the first second it ejects exhaust gas and has an acceleration of 15.6

.

Part A What is the speed Hint A.1

of the exhaust gas relative to the rocket?

How to approach the problem Hint not displayed

Hint A.2

The acceleration of the rocket Hint not displayed

Hint A.3

Find the change in mass of the rocket Hint not displayed

Express your answer numerically in kilometers per second. ANSWER:

= Answer not displayed

of its mass as

ANSWER:

= Answer not displayed

A Relation Between Momentum and Kinetic Energy Part A A cardinal (Richmondena cardinalis ) of mass 3.60×10−2

and a baseball of mass 0.144

the same kinetic energy. What is the ratio of the cardinal's magnitude

of momentum to the

of the baseball's momentum?

magnitude Hint A.1

have

How to approach the problem Hint not displayed

Hint A.2

Find a relationship between kinetic energy and momentum Hint not displayed

ANSWER:

= Answer not displayed

Part B A man weighing 720

and a woman weighing 460

the man's kinetic energy Hint B.1

to that of the woman

have the same momentum. What is the ratio of ?

How to approach the problem Hint not displayed

Hint B.2

Find a relationship between momentum and kinetic energy Hint not displayed

ANSWER:

= Answer not displayed

Collision at an Angle Two cars, both of mass north at speed

, collide and stick together. Prior to the collision, one car had been traveling

, while the second was traveling at speed

at an angle

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

south of east (as indicated at an angle

east of north.

Part A Find the speed Hint A.1

of the joined cars after the collision.

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 ANSWER:

and

.

= Answer not displayed

Part B

Part not displayed

Two Worlds on a String Two balls, A and B, with masses

and

are connected by a taut, massless string, and are moving

along a horizontal frictionless plane. The distance between the centers of the two balls is instant, the velocity of ball B has magnitude

. At a certain

and is directed perpendicular to the string and parallel to

the horizontal plane, and the velocity of ball A is zero.

Part A Find

, the tension in the string.

Hint A.1

Descibe the nature of the motion Hint not displayed

Hint A.2

The key idea Hint not displayed

Hint A.3

Find the velocity of the center of mass Hint not displayed

Hint A.4

Find the rotational speed Hint not displayed

Hint A.5

Find the radius of rotation Hint not displayed

Hint A.6

Acceleration of ball B Hint not displayed

Express ANSWER:

in terms of

,

,

, and

.

= Answer not displayed

Score Summary: Your score on this assignment is 93.8%. You received 37.5 out of a possible total of 40 points.