PC1431 MasteringPhysics Assignment 4

Assignment 4: Linear MomentumDue: 2:00am on Saturday, October 9, 2010

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A Game of Frictionless Catch

Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined massof 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 relativeto the ground after she catches the ball is .

When answering the questions in this problem, keep the following in mind:The original mass of Chuck and his cart does not include the mass of the ball.

The speed of an object is the magnitude of its velocity. An object's speed will always be a nonnegativequantity.

Part A

Find the relative speed between Chuck and the ball after Chuck has thrown the ball.

Hint A.1 How to approach the problem

Hint not displayed

Express the speed in terms of and .

ANSWER: =

Correct

Make sure you understand this result; the concept of "relative speed" is important. In general, iftwo 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. Iftwo objects are moving in the same direction, then the relative speed between them is theabsolute value of the difference of the their two speeds with respect to the ground.

Part B

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

Hint B.1 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 (relative to the ground) after he throws the ball?

Hint C.1 How to approach the problem

Hint not displayed

Express your answer in terms of , , and .

ANSWER:

=

Correct

Part D

Find Jackie's speed (relative to the ground) after she catches the ball, in terms of .

Hint D.1 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 (relative to the ground) after she catches the ball, in terms of .

Hint E.1 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 theacceleration due to gravity.

Part A

What is the speed of the girl immediately before she grabs the box?


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 of the girl immediately after she grabs the box?



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 energylost 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 with respect to the

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

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 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: 2.78Correct

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 itsmomentum is not conserved.

Part C

The drag is proportional to the square of the speed of the boat, in the form . What is the

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.

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 ofthe 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 thecar (including the fuel) is 6000 kilograms, and the mass of the fuel is one-third of the total mass of thecar (i.e., 2000 killograms). The car is moved to the starting line (and left at rest), at which time the rocketis 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 frictionin this problem.

Part A

Find the acceleration of the car just after the rocket is ignited.


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 accelerationtypically experienced by thrill seekers on a roller coaster, so the driver is in no danger on thisscore.

Part B

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

Hint B.1 What has changed?

Hint not displayed

Hint B.2 Find the final mass

Hint not displayed


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 thetrip, 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.

Hint C.1 Find the change in speed

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 , the initial mass of the car (plus

fuel) , and the final mass of the car .

ANSWER: = Answer not displayed


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. Thisproblem was based loosely on the breaking of the sound barrier by the ThrustSSC team inOctober 1997.

Three-Block Inelastic Collision

A block of mass moving with speed undergoes a completely inelastic collision with a stationary

block of mass . The blocks then move, stuck together, at speed . After a short time, the two-block

system collides inelastically with a third block, of mass , which is initially stationary. The three blocks

then move, stuck together, with speed . All

three blocks have nonzero mass. Assume thatthe blocks slide without friction.

Part A

Find , the ratio of the velocity of the two-block system after the first collision to the velocity of

the block of mass before the collision.

Hint A.1 What physical principle to use

Hint not displayed

Express your answer in terms of , , and/or .


Part B

Find , the ratio of the kinetic energy of the two-block system after the first collision to the

kinetic energy of the block of mass before the collision.

Hint B.1 Formula for kinetic energy

Hint not displayed



Part C

Find , the ratio of the velocity of the three-block system after the second collision to the velocity

of the block of mass before the collisions.

Hint C.1 Total mass of the blocks

Hint not displayed



Part D

Find , the ratio of the kinetic energy of the three-block system after the second collision to the

initial kinetic energy of the block of mass before the collisions.



Part E

Suppose a fourth block, of mass , is included in the series, so that the three-block system with

speed collides with the fourth, stationary, block. Find , the ratio of the kinetic energy of all

the blocks after the final collision to the initial kinetic energy of the block of mass before any of

the collisions.

Hint E.1 How to approach the question

Hint not displayed

Express your answer in terms of , , , and/or .


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. Theautomobiles have different masses and incoming velocities as shown. After impact, the automobiles remainjoined 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:

Answernotdisplayed

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

Hint A.1 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 in terms of .


Pucks on Ice

Two hockey players, Aaron and Brunnhilde, are pushing two pucks on a frictionless ice rink. The pucksare initially at rest on the starting line.Brunnhilde is pushing puck B, which has a massthree times as great as that of puck A, whichAaron is pushing. The players exert equalconstant forces of magnitude on their pucks,

directed horizontally, towards the finish line.They start pushing at the same time, and eachplayer pushes his or her puck until it crosses thefinish 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 thisquestion.

Answer notdisplayed

Part B

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:

You need more information to decide.

Answer not displayed

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 of its mass as

exhaust gas and has an acceleration of 15.6 .

Part A

What is the speed of the exhaust gas relative to the rocket?


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.



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 have

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

magnitude of the baseball's momentum?


Hint not displayed

Hint A.2 Find a relationship between kinetic energy and momentum

Hint not displayed


Part B

A man weighing 720 and a woman weighing 460 have the same momentum. What is the ratio of

the man's kinetic energy to that of the woman ?


Hint not displayed

Hint B.2 Find a relationship between momentum and kinetic energy

Hint not displayed


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 .


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 . At a certain

instant, the velocity of ball B has magnitude 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

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Hint A.3 Find the velocity of the center of mass

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Hint A.4 Find the rotational speed

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Hint A.5 Find the radius of rotation

Hint not displayed

Hint A.6 Acceleration of ball B

Hint not displayed

Express in terms of , , , and .


Score Summary:

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

PC1431 MasteringPhysics Assignment 4

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