Assignment 3: Work and Energy

Due: 2:00am on Friday, February 22, 2013

Note: To understand how points are awarded, read your instructor's Grading Policy.

Bungee Jumping

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass

, and the surface of the bridge is a height above the water. The bungee cord, which has length when

unstretched, will first straighten and then stretch as Kate falls.

Assume the following:

The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant .

Kate doesn't actually jump but simply steps off the edge of the bridge and falls straightdownward.Kate's height is negligible compared to the length of the bungee cord. Hence, she can betreated as a point particle.

Use for the magnitude of the acceleration due to gravity.

Part A

How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finallyto rest? Assume that she doesn't touch the water.

Express the distance in terms of quantities given in the problem introduction.

Hint 1. Decide how to approach the problem

Here are three possible methods for solving this problem:

1. No nonconservative forces are acting, so mechanical energy is conserved. SetKate's gravitational potential energy at the top of the bridge equal to the springpotential energy in the bungee cord (which depends on the cord's final length )

and solve for .

2. Since nonconservative forces are acting, mechanical energy is not conserved. Setthe spring potential energy in the bungee cord (which depends on ) equal to

Kate's gravitational potential energy plus the work done by dissipative forces.Eliminate the unknown work, and solve for .

3. When Kate comes to rest she has zero acceleration, so the net force acting onher must be zero. Set the spring force due to the bungee cord (which depends on

) equal to the force of gravity and solve for .

Which of these options is the simplest, most accurate way to find given the information

available?

ANSWER:

PC1431AY1213SEM2

Assignment 3: Work and Energy Resources

Signed in as Mikael Lemanza Help Close

Correct

Hint 2. Compute the force due to the bungee cord

When Kate is at rest, what is the magnitude of the upward force the bungee cord exerts on

her?

Express your answer in terms of the cord's final stretched length and quantities given

in the problem introduction. Your answer should not depend on Kate's mass .

Hint 1. Find the extension of the bungee cord

The upward force on Kate is due to the extension of the bungee cord. What is thisextension?

Express your answer in terms of the cord's final (stretched) length and .

ANSWER:

Correct

Hint 2. Formula for the force due to a stretched cord

The formula for the force due to a stretched cord is,

where is the spring constant of the cord and is the extension of the cord.

ANSWER:

Correct

Set this force equal to Kate's weight, and solve for .

ANSWER:

a

b

c

Extension =

=

=

Correct

Part B

If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), whatis the spring constant ? Ignore all dissipative forces.

Express in terms of , , , and .

Hint 1. Decide how to approach the problem

Here are three possible methods for solving this problem:

1. Since nonconservative forces are ignored, mechanical energy is conserved. SetKate's gravitational potential energy at the top of the bridge equal to the springpotential energy in the bungee cord at the lowest point (which depends on ) and

solve for .

2. Nonconservative forces can be ignored, so mechanical energy is conserved. Setthe spring potential energy in the bungee cord (which depends on ) equal to

Kate's gravitational potential energy at the top of the bridge plus the work done bygravity as Kate falls. Compute the work done by gravity, then solve for .

3. When Kate is being held just above the water she has zero acceleration, so thenet force acting on her must be zero. Set the spring force due to the bungee cord(which depends on ) equal to the force of gravity and solve for .

Which of these options is the simplest, most accurate way to find given the information

available?

ANSWER:

Correct

Hint 2. Find the initial gravitational potential energy

What is Kate's gravitational potential energy at the moment she steps off the bridge? (Define

the zero of gravitational potential to be at the surface of the water.)

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

ANSWER:

a

b

c

=

Correct

Hint 3. Find the elastic potential energy in the bungee cord

What is the elastic potential energy stored in the bungee cord when Kate is at the lowest

point of her first downward trip?

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

Hint 1. Formula for elastic potential energy

The elastic potential energy of the bungee cord (which we are treating as an ideal spring)is

,

where is the amount by which the cord is stretched beyond its unstretched length.

Hint 2. How much is the bungee cord stretched?

By how much is the bungee cord stretched when Kate is at a depth below the bridge?

Express your answer in terms of and .

ANSWER:

Correct

ANSWER:

Correct

ANSWER:

Correct

Dancing Balls

=

=

=

Four balls, each of mass , are connected by four identical relaxed springs with spring constant . The

balls are simultaneously given equal initial speeds directed away from the center of symmetry of the

system.

Part A

As the balls reach their maximum displacement, their kinetic energy reaches __________.

ANSWER:

Correct

Part B

Use geometry to find , the distance each of the springs has stretched from its equilibrium position. (It

may help to draw the initial and the final states of the system.)

Express your answer in terms of , the maximum displacement of each ball from its initial

position.

ANSWER:

Correct

Part C

Find the maximum displacement of any one of the balls from its initial position.

a maximum

zero

neither a maximum nor zero

=

Express in terms of some or all of the given quantities , , and .

Hint 1. A useful equation

The equation

could be useful. If you are familiar with this equation, you most likely have seen the expressionapplied to a single mass on a single spring. For the situation with four balls and four masses,you will need to consider carefully which quantities to use in this expression.

ANSWER:

Correct

± If You Don't Want to Walk to the Kitchen...

As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, infront of a large, compressed spring. The spring is compressed 5.00 from its equilibrium position, and a

glass sits 19.8 from her outstretched foot.

Part A

For what value of the spring constant does Albertine just reach the glass without knocking it over?

Determine the answer "experimentally" by playing with the applet.

Express your answer in newtons per meter.

ANSWER:

Correct

Part B

Assuming that Albertine's mass is 60.0 , what is , the coefficient of kinetic friction between the

chair and the waxed floor? Use = 9.80 for the magnitude of the acceleration due to gravity.

Assume that the value of found in Part A has three significant figures.

Note that if you did not assume that has three significant figures, it would be impossible to get three

significant figures for , since the length scale along the bottom of the applet does not allow you to

measure distances to that accuracy with different values of .

=

= 95.0

Express your answer to three significant figures.

Hint 1. How to approach the problem

The best approach is to use the work-energy theorem. Initially, Albertine has only potentialenergy due to the compressed spring. When she stops at the glass without knocking it over,she has no mechanical energy. Therefore, her initial potential energy must be equal to the workdone on her by nonconservative forces during her trip. Setting these equal, you can solve for thecoefficient of friction.

Hint 2. Find Albertine's initial potential energy

What is Albertine's initial potential energy before the spring is released?

Express your answer in joules to four significant figures.

Hint 1. Potential energy of a spring

Recall that the potential energy of a spring with spring constant is given by the

formula , where is the distance that the spring is compressed or

stretched from equilibrium.

ANSWER:

Correct

Hint 3. Find the work done by nonconservative forces

Determine the work done by nonconservative forces during Albertine's trip. Use 60.0 for

Albertine's mass, 9.80 for the magnitude of the acceleration due to gravity, for the

coefficient of kinetic friction between the chair and the floor, and 19.8 for the distance traveled.

Be careful of your signs.

Express your answer numerically in terms of .

Hint 1. Which forces act on Albertine?

The only nonconservative force acting on Albertine during her trip is friction. Gravity isignored, because it is a conservative force (and even if it were not, it acts perpendicularto her direction of motion and thus does not do work). The same can be said about thenormal force.Recall that the magnitude of the frictional force is given by the formula

, where is the magnitude of the normal force.

ANSWER:

= 1188

= −1.16×104

Correct

Note that your answer is negative (by definition, is positive) because friction opposes

motion, and any force that opposes motion does negative work.

ANSWER:

Correct

Part C

The principle of conservation of energy states that energy is neither created nor destroyed. Which ofthe following describes the transformation of energy in this problem?

ANSWER:

Correct

This applet shows how the energy transforms throughout Albertine's journey. Notice that herkinetic energy is never equal to her initial potential energy, because friction is acting even as thespring expands. Try changing the spring constant and observe how the transformation of energy isaffected.

Loop the Loop

A roller coaster car may be approximated by a block of mass . The car, which starts from rest, is

released at a height above the ground and slides along a frictionless track. The car encounters a loop of

radius , as shown. Assume that the initial height is great enough so that the car never loses contact

with the track.

= 0.102

Conservation of energy does not apply to problems involving nonconservative forces. Thus,the potential energy slowly disappears during Albertine's trip.

The potential energy was turned into Albertine's kinetic energy, which was then convertedinto internal (thermal) energy.

The potential energy was turned into Albertine's kinetic energy, which is now stored in thefloor as frictional potential energy.

The potential energy was turned into elastic frictional energy, creating the frictional force.

Part A

Find an expression for the kinetic energy of the car at the top of the loop.

Express the kinetic energy in terms of , , , and .

Hint 1. Find the potential energy at the top of the loop

What is the potential energy of the car when it is at the top of the loop? Define the gravitationalpotential energy to be zero at .

Express your answer in terms of and other given quantities.

ANSWER:

Correct

ANSWER:

Correct

Part B

Find the minimum initial height at which the car can be released that still allows the car to stay in

contact with the track at the top of the loop.

Express the minimum height in terms of .

=

=

Hint 1. How to approach this part

Meaning of "stay in contact"

For the car to just stay in contact through the loop, without falling, the normal force that acts onthe car when it's at the top of the loop must be zero (i.e., ).

Find the velocity at the top such that the remaining force on the car i.e. its weight provides thenecessary centripetal acceleration. If the velocity were any greater, you would additionallyrequire some force from the track to provide the necessary centripetal acceleration. If thevelocity were any less, the car would fall off the track.

Use the above described condition to find the velocity and then the result from the above part tofind the required height.

Hint 2. Acceleration at the top of the loop

Assuming that the speed of the car at the top of the loop is , and that the car stays on the

track, find the acceleration of the car. Take the positive y direction to be upward.

Express your answer in terms of and any other quantities given in the problem

introduction.

ANSWER:

Correct

Hint 3. Normal force at the top of the loop

Suppose the car stays on the track and has speed at the top of the loop. Use Newton's

2nd law to find an expression for , the magnitude of the normal force that the loop exerts on

the car when the car is at the top of the loop.

Express your answer in terms of , , , and .

Hint 1. Find the sum of forces at the top of the loop

Find the sum of the forces acting on the car at the top of the loop. Remember that thepositive y direction is upward.

Express your answer in terms of , , and .

ANSWER:

Correct

ANSWER:

=

=

Correct

Hint 4. Solving for

The requirement to stay in contact results in an expression for in terms of and .

Substitute this into your expression for kinetic energy, found in Part A, to determine a relationbetween and .

ANSWER:

Correct

For the car will still complete the loop, though it will require some normal reaction even

at the very top.

For the car will just oscillate. Do you see this?

For , the cart will lose contact with the track at some earlier point. That is why

roller coasters must have a lot of safety features. If you like, you can check that the angle at

which the cart loses contact with the track is given by . Where is the

angle measured counterclockwise from the horizontal positive x-axis, where the origin of the x-axisis at the center of the loop.

Shooting a ball into a box

Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is

fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height

above the top of the box (the height of which is negligibly small), and the center of the box is a distance

from the edge of the table. The spring has a spring constant . The first child compresses the spring a

distance and finds that the marble falls short of its target by a horizontal distance .

=

=

Part A

By what distance, , should the second child compress the spring so that the marble lands in the

middle of the box? (Assume that height of the box is negligible, so that there is no chance that themarble will hit the side of the box before it lands in the bottom.)

Express the distance in terms of , , , , and .

Hint 1. General method for finding

For this part of the problem, you don't need to consider the first child's toss. (The quantities

and should not appear in your answer.) Consider the energy conservation and kinematic

relations for the marble, and solve for its range, , in terms of , , , and .

Hint 2. Initial speed of the marble

Use conservation of energy to find the initial speed, , of the second marble.

Express your answer in terms of , , and .

ANSWER:

Correct

Hint 3. Time for the marble to hit the ground

Use kinematics to find , the time it takes the second marble to hit the ground after it is shot

off the table.

Express your answer in terms of and .

ANSWER:

=

Correct

Hint 4. Combining equations and solving for

The kinematic equation for the motion along the x axis is . Using the expressions for

and from the previous hints, solve for in terms of the quantities , , , , and .

ANSWER:

Correct

Part B

Now imagine that the second child does not know the mass of the marble, the height of the table abovethe floor, or the spring constant. Find an expression for that depends only on and distance

measurements.

Express in terms of , , and .

Hint 1. Compute

Use your answer to Part A to write in terms of , , , , , and .

ANSWER:

Correct

So it is just like the first case, with replacing . Now divide the equations for

and by each other.

ANSWER:

=

=

=

Correct

Work Raising an Elevator

Look at this applet. It shows an elevator with a small initial upward velocity being raised by a cable. Thetension in the cable is constant. The energy bar graphs are marked in intervals of 600 .

Part A

What is the mass of the elevator? Use for the magnitude of the acceleration of gravity.

Express your answer in kilograms to two significant figures.

Hint 1. Using the graphs

Think about which graph(s) show energies that are directly related to the mass of the elevator.There may be more than one. You would like to get the most accurate number you can, sochoose the graph that you can read most accurately.

Hint 2. Needed formula

Recall that the gravitational potential energy near the earth's surface is given by ,

where is the mass of the object, is the magnitude of the gravitational acceleration, and

is the height above the ground.

ANSWER:

Correct

Part B

Find the magnitude of the tension in the cable. Be certain that the method you are using will be

accurate to two significant figures.

Express your answer in newtons to two significant figures.

Hint 1. How to approach the problem

In the previous part, you could use the graph of potential energy to determine the mass to twosignificant figures, because when the elevator stopped, the top of the potential energy bar layright on one of the grid lines. In this problem, you could use the graph of work to find thetension, but since it lies somewhere between the grid lines, it is unlikely that you coulddetermine the tension to the necessary accuracy. However, it is a good way to get an estimate

=

= 60

with which to check your answer.

The numerical data given in the window beneath the graphs do have two significant figures ofaccuracy, and thus they could be used in combination with the data in the graph of the finalenergy to get a more accurate value for the work done on the elevator. Recall, in fact, that thework done on the elevator by the tension must equal the change in mechanical energy of thesystem.

Hint 2. Find the change in mechanical energy

From the information given in the applet and the information found in Part A, determine thechange in the total mechanical energy of the system .

Express your answer in joules to two significant figures.

Hint 1. Find the initial mechanical energy

Assuming that the potential energy of the elevator at the instant when you run thesimulation is zero, what is the initial mechanical energy of the system?

Express your answer in joules to two significant figures.

Hint 1. Definition of mechanical energy

Recall that the mechanical energy of a system is defined as the sum of kineticenergy and potential energy,

.

Note that, at the instant when you run the simulation, the potential energy of

the elevator is zero. Thus, the total initial mechanical energy of the system issimply given by the initial kinetic energy of the elevator , which

can be evaluated from the information about the mass of the elevator found in PartA, and the information about the initial speed of the elevator given in the windowbeneath the bar graphs in the applet.

ANSWER:

Correct

The total mechanical energy of the system can be determined from the data in theenergy bar graphs given in the applet, just as you did in Part A to find the mass ofthe elevator.

ANSWER:

= 480

= 1900

Correct

Since the change in the total mechanical energy of the system must equal the work

done by the tension, your answer gives a more accurate estimate of than what you

could have calculated from the data in the work bar graphs in the applet. Now use theinformation about the distance moved by the elevator given in the window beneath thegraphs to find the tension.

ANSWER:

Correct

Power Dissipation Puts a Drag on Racing

The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds is calledform drag. It increases quadratically with velocity (essentially because the amount of air you run intoincreases with and so does the amount of force you must exert on each small volume of air). Thus

,

where is the cross-sectional area of the vehicle and is called the coefficient of drag.

Part A

Consider a vehicle moving with constant velocity . Find the power dissipated by form drag.

Express your answer in terms of , , and speed .

Hint 1. How to approach the problem

Because the velocity of the car is constant, the drag force is also constant. Therefore, you can

use the result that the power provided by a constant force to an object moving with

constant velocity is . Be careful to consider the relative direction of the drag force

and the velocity.

ANSWER:

Correct

Part B

= 480

=

A certain car has an engine that provides a maximum power . Suppose that the maximum speed of

the car, , is limited by a drag force proportional to the square of the speed (as in the previous part).

The car engine is now modified, so that the new power is 10 percent greater than the original power

( .

Assume the following:

The top speed is limited by air drag.The magnitude of the force of air drag at these speeds is proportional to the square of thespeed.

By what percentage, , is the top speed of the car increased?

Express the percent increase in top speed numerically to two significant figures.

Hint 1. Find the relationship between speed and power

If the magnitude of the air-drag force is proportional to the square of the car's speed, how is thepower delivered, , related to the speed ?

ANSWER:

Correct

Hint 2. How is the algebra done?

The relationship between the new power and the old power is . The relationship

between the new top speed and the old top speed can be written as , where

is the percent change in top speed. Finally, power is related to maximum speed by the

formula .

What is in terms of ?

Hint 1. Help with some math

Starting with the relationship,

substitute in the expressions for and in terms of and :

.

Then, divide this last expression by the relationship

.

This is a general approach to scaling problems. The advantage is that the unknownconstant of proportionality (in this case ) divides out.

ANSWER:

Correct

ANSWER:

Correct

You'll note that your answer is very close to one-third of the percentage by which the power wasincreased. This dependence of small changes on each other, when the quantities are related byproportionalities of exponents, is common in physics and often makes a useful shortcut forestimations.

Pulling a Block on an Incline with Friction

A block of weight sits on an inclined plane as shown. A force of magnitude is applied to pull the

block up the incline at constant speed. Thecoefficient of kinetic friction between the planeand the block is .

Part A

= 3.2 %

What is the total work done on the block by the force of friction as the block moves a distance

up the incline?

Express the work done by friction in terms of any or all of the variables , , , , , and .

Hint 1. How to start

Draw a free-body force diagram showing all real forces acting on the block.

Hint 2. Find the magnitude of the friction force

Write an expression for the magnitude of the friction force.

Express your answer in terms of any or all of the variables , , , and .

Hint 1. Find the magnitude of the normal force

What is the magnitude of the normal force?

Express your answer in terms of , , and .

ANSWER:

Correct

ANSWER:

Correct

ANSWER:

Correct

Part B

What is the total work done on the block by the applied force as the block moves a distance

up the incline?

Express your answer in terms of any or all of the variables , , , , , and .

ANSWER:

=

=

=

Correct

Now the applied force is changed so that instead of pulling the block up the incline, the force pulls the blockdown the incline at a constant speed.

Part C

What is the total work done on the block by the force of friction as the block moves a distance

down the incline?

Express your answer in terms of any or all of the variables , , , , , and .

ANSWER:

Correct

Part D

What is the total work done on the box by the appled force in this case?

Express your answer in terms of any or all of the variables , , , , , and .

ANSWER:

Correct

Dragging a Board

=

=

=

A uniform board of length and mass lies near a boundary that separates two regions. In region 1, the

coefficient of kinetic friction between the board and the surface is , and in region 2, the coefficient is .

The positive direction is shown in the figure.

Part A

Find the net work done by friction in pulling the board directly from region 1 to region 2. Assume

that the board moves at constant velocity.

Express the net work in terms of , , , , and .

Hint 1. The net force of friction

Suppose that the right edge of the board is a distance from the boundary, as shown. When

the board is at this position, what isthe magnitude of the force of friction,

, acting on the board

(assuming that it's moving)?

Express the force acting on theboard in terms of , , , , ,

and .

Hint 1. Fraction of board in region 2

Consider the part of the board in region 2 when the right edge of the board is a distance from the boundary. The magnitude of the force of friction acting on the board (only

considering the friction from region 2) will be the coefficient of friction, multiplied by themagnitude of the normal force that acts on the board. Since the ground is horizontal, andthe board is not accelerating in the vertical direction, the normal force should equal the

board's weight. But, only a fraction of the board's total mass is in region 2. Find thefraction of the board in region 2 in terms of the given lengths;

.

ANSWER:

Correct

Hint 2. Force of friction in region 1

Now consider that part of the board in region 1. Again, only a fraction of the board's massis in region 1. Using this fact, find the magnitude of the force of friction acting on theboard, just due to friction in region 1.

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

Hint 1. Fraction of the board in region 1

When the right edge of the board is a distance from the boundary, what fraction

of the board lies in region 1?

.

ANSWER:

Correct

ANSWER:

Correct

ANSWER:

Correct

Fraction of board in region 2 =

Fraction of board in region 1 =

=

=

Hint 2. Work as integral of force

After you find the net force of friction that acts on the board, as a function of , to find the

net work done by this force, you will need to perform the appropriate work integral,

The lower limit of this integral will be at . What will be the upper limit?

ANSWER:

Correct

Hint 3. Direction of force of friction

Don't forget that the force of friction is directed opposite to the direction of the board's motion.

Hint 4. Formula for

ANSWER:

Correct

This answer makes sense because it is as if the board spent half its time in region 1, and half inregion 2, which on average, it in fact did.

Part B

What is the total work done by the external force in pulling the board from region 1 to region 2? (Again,assume that the board moves at constant velocity.)

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

Hint 1. No acceleration

Since the board is not accelerating, the sum of the external forces on it must be zero. Thereforethe external force must be oppositely directed to that of friction.

ANSWER:

Upper limit at =

=

Correct

Circling Ball

A ball of mass is attached to a string of length . It is being swung in a vertical circle with enough speed

so that the string remains taut throughout the ball's motion. Assume that the ball travels freely in thisvertical circle with negligible loss of totalmechanical energy. At the top and bottom of thevertical circle, the ball's speeds are and ,

and the corresponding tensions in the string are

and . and have magnitudes and

.

Part A

Find , the difference between the magnitude of the tension in the string at the bottom relative to

that at the top of the circle.

Express the difference in tension in terms of and . The quantities and should not

appear in your final answer.

Hint 1. How to approach this problem

Identify the forces that act on the ball as it moves along the circular path. Then, write equationsfor the sum of the forces on the ball at the top and the bottom of the path. Next, use Newton'ssecond law to relate these net forces to the acceleration of the ball. Notice that the ball doesnot move with uniform speed so the acceleration of the ball at the top of the circle is differentfrom the acceleration at the bottom of the circle.To finish the problem, you may want to use energy conservation to relate the speed of the ballat the bottom of the circle to the speed at the top.

Hint 2. Find the sum of forces at the bottom of the circle

What is the magnitude of the net force in the y direction acting on the ball at the bottom of thecircle?

Express your answer in terms of the variables given in the problem. You may use to

represent the acceleration of gravity, 9.8 .

=

ANSWER:

Correct

Hint 3. Find the acceleration at the bottom of the circle

Find , the magnitude of the vertical acceleration of the ball at the bottom of its circle.

Express your answer in terms of and possibly other given quantities.

ANSWER:

Correct

Hint 4. Find the tension at the bottom of the circle

Find the magnitude of the tension in the string when the ball is at the bottom of the circle.

Express your answer in terms of , , , and the speed of the ball at the bottom of

the circle.

Hint 1. What physical principle to use

Apply Newton's 2nd law in the y direction to obtain .

ANSWER:

Correct

Hint 5. Find the sum of forces at the top of the circle

What is the magnitude of the net force in the y direction acting on the ball at the top of itscircle?

Express your answer in terms of the variables given in the problem. You may use to

represent the acceleration of gravity, 9.8 .

ANSWER:

=

=

=

Correct

Hint 6. Find the acceleration at the top of the circle

Find , the magnitude of the vertical acceleration of the ball at the top of its circle.

Express your answer in terms of and possibly other given quantities.

ANSWER:

Correct

Hint 7. Find the tension at the top of the circle

Find the magnitude of the tension in the string when the ball is at the top of the circle.

Express your answer in terms of , , , and the speed of the ball at the top of the

circle.

Hint 1. Relationship to solution for

Follow the same steps you used to find (see Hint 3), noting carefully where various

directions (signs) are reversed.

ANSWER:

Correct

Hint 8. Find the relationship between and

The total mechanical energy of the system is the same when the ball is at the top and bottom ofthe vertical circle. Use conservation of energy to write an expression for in terms of .

Your answer may also include , , and .

ANSWER:

=

=

=

=

Correct

ANSWER:

Correct

The method outlined in the hints is really the only practical way to do this problem. If doneproperly, finding the difference between the tensions, , can be accomplished fairly simply

and elegantly.

Drag on a Skydiver

A skydiver of mass jumps from a hot air balloon and falls a distance before reaching a terminal

velocity of magnitude . Assume that the magnitude of the acceleration due to gravity is .

Part A

What is the work done on the skydiver, over the distance , by the drag force of the air?

Express the work in terms of , , , and the magnitude of the acceleration due to gravity .

Hint 1. How to approach the problem

If no nonconservative forces were acting, then the total mechanical energy (kinetic pluspotential) of the skydiver upon leaving the plane would be equal to the total mechanical energyof the skydiver after falling a distance .

Now consider the drag force, which is nonconservative. The drag force opposes the motion ofthe skydiver, which means that it does negative work on the skydiver. Thus, the final mechanicalenergy of the skydiver will be smaller than the initial mechanical energy by an amount equal tothe work done by the drag force.

Hint 2. Find the change in potential energy

Find the change in the skydiver's gravitational potential energy, after falling a distance .

Express your answer in terms of given quantities.

ANSWER:

Correct

Hint 3. Find the change in kinetic energy

=

=

Find the change in the skydiver's kinetic energy, after falling a distance .

Express your answer in terms of given quantities.

ANSWER:

Correct

ANSWER:

Correct

Part B

Find the power supplied by the drag force after the skydiver has reached terminal velocity .

Express the power in terms of quantities given in the problem introduction.

Hint 1. How to approach the problem

One way to approach this problem would be to apply the definition of power as the timederivative of the work done. A simpler method that works for this problem is to use the formula

for the power delivered by a force acting on an object moving with velocity :

.

Hint 2. Magnitude of the drag force

Find the magnitude of the drag force after the skydiver has reached terminal velocity.

Express your answer in terms of the skydiver's mass m and other given quantities.

Hint 1. Definition of terminal velocity

Once terminal velocity is reached, the skydiver's acceleration goes to zero. (The dragforce exactly balances the downward acceleration due to gravity.)

ANSWER:

Correct

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=

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Hint 3. Relative direction of the drag force and velocity

When you find , remember that the direction of the drag force is opposite to the direction

of the skydiver's velocity.

ANSWER:

Correct

Score Summary:

Your score on this assignment is 99.8%.You received 49.9 out of a possible total of 50 points.

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