PC1431 MasteringPhysics Assignment 4

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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:

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 nonnegative quantity.

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, 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 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

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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

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

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

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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 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

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?

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Hint not displayed Hint B.2 Total initial momentum

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 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

Express your answer numerically in meters. ANSWER:

= 2.81

Correct

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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.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 . 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 C.2 Find the time rate of change of momentum of the boat Hint not displayed

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

Hint B.1 What has changed?

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Hint B.2 Find the final mass

= 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.

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}

= 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 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 that the blocks slide without friction.

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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

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

= _{Answer not displayed}

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

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

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ANSWER: ₌

Answer 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

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

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

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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.

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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

Express in terms of . ANSWER: ₌

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.

Answer not displayed

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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.

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

Express your answer numerically in kilometers per second.

ANSWER: ₌

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ANSWER: ₌

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 A.2 Find a relationship between kinetic energy and momentum Hint not displayed

ANSWER:

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 B.2 Find a relationship between momentum and kinetic energy Hint not displayed

ANSWER:

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.

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Part A

Find the speed of the joined cars after the collision.

Hint A.1 Determine the conserved quantities

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

Express your answer in terms of and . ANSWER: ₌ _{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 . 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.

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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

Express in terms of , , , and . ANSWER: ₌

Score Summary:

Your score on this assignment is 93.8%.