Work, Energy, and Power

Chapter 6

Energy is a science term that has been part of the political landscape across the world for many years. Energy makes our homes warm or cool, drives our cars and trains, and makes all our wonderful electrical devices operate. Lots of energy comes from the Sun, and even fossil fuels and wind power get their origins from the Sun. In terms of describing nature, energy will become the most powerful concept in all of physics. And yet, we do not understand it. Richard Feynman, who was a famous physicist who developed much of our understanding of modern physics, said that “it is important to realize that in physics today, we have no knowledge of what energy is.” This level of the unknown makes physics an adventure. The lessons contained in this chapter will have some broad applications, with a touch of abstraction.

Work and Energy – Lesson 1

Lesson Objectives

• Define work done by various forces

• Determine work done by variable forces

• Explain the relationship between work and energy

• Understand the concept of power

Vocabulary

Work

Energy

Power

Check Your Understanding

1. If a force is applied to an object over some distance, what are some of the possible results for the motion of that object?

2. How would you use the words work and energy to describe what happens when 50 concrete blocks are moved from downstairs to upstairs?

Introduction

The use of the words work and energy in everyday conversation creates meanings that are not recognized by their definitions in physics. Someone sitting at their desk all day may say that they have done a lot of work, while physics would says they have done very little. It is incorrect to pay a “power bill” in terms of the physics definition of energy and power. These concepts are much more “big picture” than the concepts of motion and forces. Despite the increased realm of these new concepts, many problems can be solved more simply than they were with motion and forces. To take on these concepts, it is important to embrace the physics definition of both and to let go of the fuzzy translations used in casual discussions.

Lesson Content

Defining Work

Work is defined very specifically in physics as a quantifiable effort that is the result of a force causing a displacement. These words can be described with the following equation:

W F x= ⋅∆

.

Work is a dot product, and thus is not a vector. The units of work are Newton-seconds, which have been renamed Joules after Thermodynamics experimenter James Joule. A dot product can be written in simpler terms with the cosine of the angle between the two vectors, so that work can be expressed as:

/cos F xW F x θ ∆= ∆

where θF/∆x is the angle between the force and the displacement

Consider the implications of this definition. If you hold a box of books at arm’s length for several minutes, your muscles will quickly tire and the sensation of pain from built up lactic acid will suggest that you are doing a lot of work. Since ∆x = 0, you would have done no work. Now, take that same box of books at arm’s length and walk 150m. The same pain would be felt, and still no work will be done. Clearly the applied force and the displacement were not zero. Figure 1 shows the importance of the dot product and its cosine function. On the left, the angle between the red force and blue displacement is 90 degrees, and the cosine of 90 degrees is zero. Therefore, no work was done. This should make sense. None of the red force helped the blue displacement happen. Work is the measure of an effort that creates a displacement. The cosine from the dot product determines the component of the force that in the direction of the displacement assuming the angle is the one between the two vectors.

Figure 1. Importance of the angle between force and displacement in Work

If a force is applied in the opposite direction of an object’s motion, the resulting work will be negative. Consider the work done by the brakes on a car. The car moves to the right so the displacement is to the right. The braking force is to the left. The angle between the two is 180 degrees, and the ( )cos 180 1= − . Therefore, work done by a force resisting a motion is

negative.

Work done by variable forces

These definitions of work assume that the force applied to an object is constant throughout the application over the distance ∆x. When the force varies, the definition can be applied to a very small distance dx so that the little bit of work, dW, can be found using:

dW F dr= ⋅

The variable x is replaced with a generic displacement r that could be anywhere in three dimensions. Then, it should be clear that integration can be used to add up all the little amounts of work to find the total amount of work done:

0

r

rW dW F dr= = ⋅∫ ∫

Another powerful feature of work is that it is path independent. This can be best illustrated in an example.

Example

How much work is done by a variable force, 2ˆ ˆ3 2F xx y y= +

that moves from a position

(3.0m,2.0m) to (6.0m,5.0m)?

Solution:

Note that the question did not specify the path to be taken, only the beginning and ending positions. Figure 2 shows an arbitrary path between the initial and final positions. Since the work is a dot product of force and displacement, the actual path will not change the work quantity. The sum of the incremental works, dW, will add to be the same amount of total work no matter what path is used to get from the initial to final positions.

Figure 2. Path Independence of Work

The path can be broken into small displacements that can be expressed as:

ˆ ˆdx

dr dxx dyydy

= + =

The definition of work with a variable force is given as: 0

r

rW dW F dr= = ⋅∫ ∫

.

Replacing the vector definition of force and for the incremental displacement,

( ) 6.0 5.0 6.0 5.02 2 2 33 22 32 3.0 2.03.0 2.0

33 2 3 2

2m m m m

m mm m

x dxW xdx y dy xdx y dy x y

y dy

= ⋅ = + = + = + ∫ ∫ ∫ ∫

W = 118.5J

Defining the concept of energy

A wagon loaded with boxes of physics books is stationary next to you. If you applied a force over a distance, you could do work on the wagon of physics books. Not doing anything means that no work is done. The fact that work could be done is called energy. Energy is the ability to do work. Another way to relate work to energy is to say that work is equal to a change in energy: W E= ∆ . The units of energy are the same as work, Joules.

There are many categories of ways that potential work could be done. The next lesson will explore mechanical energy, which consists of potential and kinetic energies. Springs and rubber bands can store energy. A stretched rubber band is not doing work on anything. However, if it were released and were to push on a small ball bearing, it could become a sling shot by doing work on the bearing. Batteries can do work, so they contain energy. This is a category of chemical energy where the work is done when outer shell electrons of particular molecules interact. Electrons can also be stored in capacitors and released to flow through a circuit to do some work.

Electrical Energy involving the outer parts of atoms where electrons are located

Hydrogen bomb -- an uncontrolled release of nuclear energy in the nucleus of atoms

Figure 3. Forms of Energy

The insides of atoms can be rearranged to release one of two kinds of nuclear energy. Unstable atoms whose neutrons get isolated from the protons that keep them stable will disappear, leaving behind a proton, an electron, an anti-neutrino, and a burst of high energy radiation. This is the slow release of energy generally called radiation, and it is the reason why the core of the Earth and other planets are still molten after 4.5 billion years. A much quicker release of nuclear energy that can do incredible amounts of work happens when the nuclei are fused or split. Nuclear fission releases energy by converting some of the mass into energy

through Einstein’s famous discovery that 2.E mc= A much more efficient process of energy release happens when smaller atoms are pushed together so that they form larger ones. This process of nuclear fusion is how the Sun creates it energy that ends up doing a lot of work here on Earth and has even stored much of that potential work or energy in the form of fossil fuels and animal and plant life. These topics on the physics of atoms will be explored in greater detail in chapters 24-26.

Energy as a concept is definitely abstract, as it is not completely understood. Part of that abstraction also allows for a big picture understanding of nature without having to know all the small details. For instance, it is clear that one form of energy can be converted to another. Mechanical energy can be converted into chemical energy by turning a generator to charge a battery. This knowledge can be used to make predictions about future work to be done without knowing the many details in that process.

Power

Power is sometimes used to describe influence or potential ability to change something. The physics definition is much simpler. Power is the rate at which work can be done.

dWPdt

= and in cases where Work is done at a constant rate, WPt

= where W is work done

and t is the total time for the work to be completed.

Lesson Summary

• Work is the measure of an applied force and the displacement it causes.

• Simple work is defined by the equation, /cos F xW F x θ ∆= ∆

. It is measured in Joules.

• To find the work done by variable forces, calculus must be used: 0

r

rW dW F dr= = ⋅∫ ∫

.

• Energy is the potential to do work, and W E= ∆ .

• Energy comes in many forms and can be converted from one to the other, such as mechanical energy, chemical energy, electrical energy, and nuclear energy.

• Power is the rate that work can be done, or dWPdt

=

Review Questions

Review Problems

Further Reading / Supplemental Links

•

Points to Consider

1. Will a moving object do work on a stationary object if they collide?

2. Does a box of physics books held over your head have the potential to do work?

Mechanical Energy – Lesson 2

Lesson Objectives

• Understand the concept of gravitational potential energy

• Find the potential energy of a spring

• Know the meaning of kinetic energy

• Know how to find total mechanical energy

Vocabulary

• Gravitational potential energy

• Energy

• Power

Check Your Understanding

1. How is the definition of work in physics different than use in conversation?

2. When is calculus needed to calculate work done?

3. Why is “power bill” a bad label?

Introduction

The previous lesson discussed the forms of energy. Many of those forms are very important and influence everyday life. For a great part of your exploration of Physics World, only mechanical energy will be considered. This is the most obvious form of energy, and one that may be the easiest to visualize. Since energy is an abstract concept, this clarity plays to the benefit someone trying to master the subject and begin trying to apply it.

Lesson Content

Gravitational Potential Energy

Think about that box of books from Lesson 1 again. In defining work, the box of books was held out at arm’s length. Despite the pain it would cause, no work was done. This time, apply a force so that the box of book is elevated from the floor to over your head. The minimum force required to lift a box of books is equal to its weight, with a little extra to get it started and a little extra to stop it. This extra force will create a positive and negative work that will negate each other. The amount of work needed to lift a box of books with mass m an upward

displacement of h is equal to /cos F xW F x mghθ ∆= ∆ =

. The angle between a force applied

upward and a displacement upward is zero degrees, and ( )cos 0 1= . Now, what is the box is

released and allowed to land on your foot. Would work be done on your foot when the box of books lands on it? The box would apply a force to your foot. It would be squished a slight amount by the box of books, and this displacement would be caused by the work done on your foot by the books in the box. In this sense, a box held in the air is work waiting to happen or potential work. This concept of potential work is called energy. In particular, this type of potential work or energy is called gravitational potential energy.

For objects near the surface of the Earth where the acceleration due to gravity is mostly a constant, the gravitational potential energy can always be expressed as:

PE mgh=

The distance h is always measured from some pre-selected reference height and the resulting potential energy is the amount of work that could be done compared to this height. Typically, this height is chosen to be some ideal lowest point.

Potential Energy of a Spring

The force of a spring depends on the material and design of the spring and the amount that it is stretched. For simple springs, the force can be described as: ˆspringF k xx= − ∆

.

Figure 4. Spring types and automobile suspension applications

How much work would it take to compress a spring by an amount ∆x?

( )0

2120 0

ˆ ˆr x x

rW dW F dr kxx dxx kx dx k x

∆ ∆= = ⋅ = ⋅ = = ∆∫ ∫ ∫ ∫

This measure of work will also be energy stored in the spring since W E= ∆ .

The potential energy stored in a spring can be written: 212springPE k x= ∆ .

Energy of Motion

As a thought example, you decide to stand in front of a moving truck with the following reasoning. The truck is moving at a constant speed of 20m/s (about 44 miles per hour), which means it has a zero acceleration. Therefore, the net force on the truck is zero and it will not affect you. This line of reasoning is missing some physics (and some common sense). There must be energy stored in the moving truck because it will most certainly apply a force to you over some distance if it hits you thus doing work on you. Imagine a slow motion video of an the impact. The front of the truck hits you and slams against your body making it deform and contort until you are pinned against the front of the moving truck. This all takes place in a fraction of a second. However, the force and distance are real and so is the work done.

/ /truck person truck person person personW F x m a x= ∆ = ∆

Remember kinematics? 2 20 2v v a x= + ∆ . Solving for a x∆ gives: ( )2 21

02a x v v∆ = −

( )2 2 21 1/ 02 2truck person person personW m v v m v= − = , since the initial velocity of the person was zero.

So, a moving truck could do this much work to you. This is the energy of motion, called kinetic energy. Kinetic energy can be expressed as:

212KE mv=

Total Mechanical Energy

The total energy of a system is equal to the sum of all of the energies it has. If you are asked to find total mechanical energy, it will always be the sum of the potential and kinetic energies contained in that system. A system may be one object or it may be a group of objects. This is a choice of the person performing the analysis. Total mechanical energy will be represented as ME, Etotal, or just E since only mechanical energy is being considered right now.

Total Mechanical Energy at a particular instance in time is:

E PE KE= +∑ ∑

Lesson Summary

• Gravitational potential energy is measured from some reference level and can be expressed as, PE mgh= .

• Springs that are compressed or stretched have the ability to do work, and therefore have a potential energy written as, 21

2springPE k x= ∆ .

• Moving energy is called kinetic energy: 212KE mv= .

• Total mechanical energy is the sum of all the potential and kinetic energy of an object or system of objects.

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. A person contains energy. Is there a limit to how much work a person can do?

2. Can friction do work?

Conservation of Energy – Lesson 3

Lesson Objectives

• Define the conservation of energy

• Apply the conservation of energy when work does not sneak energy away

• Use conservation of energy when work is done by a non-conservative force

Vocabulary

Law of Conservation of Energy

Non-conservative force

Check Your Understanding

1. Describe a roller coaster that moves at rest at the top of a hill to moving fast at the bottom of a hill in terms of energy.

2. A spring loaded toy gun is held 1.0m above the floor. How do you find the total energy of the toy gun system?

Introduction

The amount of work you can do is limited by how much energy you have. This statement may seem to explain the relationship between work and energy and introduce the principle of conservation of energy. That is, until you realize that energy is defined as the ability to do work. Energy is a macroscopic definition of a microscopic world, and so it is considered abstract. It was not until the 1840’s that experiments in heat revealed a startling relationship between energy at different times. This principle is one of the most powerful tools of discovery in all of physics. Successful application of the conservation of energy can make analysis of a seemingly complex situation simple.

Lesson Content

Defining the Conservation of Energy

It was the work of James Joule and Julius Robert von Mayer that found that the amount of energy in a closed system stayed the same. One experiment by James Joule involved a hanging weight connected to a vertical rod attached to paddles immersed in water. As the weight moved downward, it turned the paddles and agitated the water. Joule showed that the loss of gravitational potential energy of the weight was equal to the gain of thermal energy of the water and paddles through friction with the water. The thermal energy was measured with

a thermometer which showed the temperature rise as the water was swirled around. Figure 5 shows a diagram of the device used by Joule.

Figure 5. The Joule Apparatus which helped establish that energy is conserved. http://en.wikipedia.org/wiki/File:Joule%27s_Apparatus_%28Harper%27s_Scan%29.png

The Law of Conservation of Energy states that the amount of total energy in a system isolated from other systems will be constant over time. It also says energy can neither be created or destroyed.

Applying the Conservation of Energy to isolated systems

In cases where you only consider mechanical energy, the conservation of energy could be written several ways.

0KE PE∆ + ∆ = or 0 1 2E E E= = =where E is total mechanical energy, n n nE KE PE= + .

To apply these conditions, there must be two instances in time to compare. The choice of these places in time is usually made based on where information is known and where you want to know information. Buried in the expressions for kinetic and potential energy are position and velocity. In many instances, the mass is on both sides of the equation and will cancel out. This is a new method of problem solving. Just as with calculations of work, the method is path independent. Energy only cares about where you started and where you ended.

Energy Method without Work done by outside forces

1. Identify your system. It may be one object or more than one object is they interact with each other. Interactions could be objects connected by ropes or, as chapter 7 will discuss, when they collide with each other.

2. Choose two instances in time to compare. These two times are chosen based on where you were able to obtain information about velocity or vertical position.

3. Apply 0KE PE∆ + ∆ = or 0 1 2E E E= = =

Example: Consider the simple example of a rock dropped from rest at a height h. Use the conservation of energy to find the velocity after falling a distance h.

Solution:

Using 0KE PE∆ + ∆ = 21

1 0 12 0KE KE KE mv∆ = − = −

1 0 0PE PE PE mgh∆ = − = −

Putting these back in the

expression, 21

12 0mv mgh− =

Solving for unknowns:

1 2v gh=

Using 0 1 2E E E= = =

( )210 0 0 2 0E PE KE mgh m= + = +

( ) 211 1 1 120E PE KE mg mv= + = +

Setting these two energies

equal to each other and solving

for the unknown velocity yields

the same results as the left. 21

12mgh mv=

1 2v gh=

Accounting for energy that leaves one system for another

In real experiments, it is a great challenge to isolate a system completely from the outside world. Energy can easily escape to the surrounding air and can even be transmitted through empty space by photons as with the energy from the Sun. For perfect systems, it can be said that the total change in energy will be zero or 0E∆ = . For imperfect systems, energy can leave the system by doing work on the outside systems or can enter the system if an outside system does work on the one being studied. This can be expressed asW E= ∆ , and applied to any situation where conservation of energy can be applied. Using this will generally be called the energy method.

Energy Method with Work done by outside forces

1. Identify your system. It may be one object or more than one object is they interact with each other. Interactions could be objects connected by ropes or as chapter 7 will discuss, when they collide with each other.

2. Choose two instances in time to compare. These two times are chosen based on where you were able to obtain information about velocity or vertical position.

3. Apply ncKE PE W∆ + ∆ = or 1 0ncW E E= − . Wnc means work done by a non-

conservative force. This includes forces that allow total energy to be changed by adding or removing energy that is not tracked with mechanical energies (potential or kinetic). The best example of this kind of force will be friction force. Friction force will take

energy away mostly in the form of hear. An infrared camera will show a slight increase in temperature if you drag a box across a carpeted floor.

Example: A ball is released from rest at the top of a ramp that is a height h above the carpeted floor. The ramp itself is frictionless, but the carpet makes the ball come to rest after rolling a distance D. Calculate the work done by friction using the energy method.

Solution:

Start withW E= ∆ . The ball starts with

some amount of potential energy.

After rolling a distance D, it has no

energy at all (potential and kinetic are

both zero). The loss of energy happened

because friction did work to remove

that energy, probably in the form of

heat and some deformation. So, work

can be found by calculating the change

in energy.

2 0 0W E E E mgh= ∆ = − = − . The work done by friction is: frictionW mgh= − . The

negative sign signifies that energy was taken out of the system.

Lesson Summary

• Energy cannot be created or destroyed. The amount of energy is an isolated system that stays constant.

• The energy method when work is NOT done by a non-conservative force can be expressed as: 0KE PE∆ + ∆ = or 0 1 2E E E= = = where n n nE KE PE= + .

• The energy method when work is done by a non-conservative force uses:

ncKE PE W∆ + ∆ = or 1 0ncW E E= − .

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. What happens to the total energy of a pendulum that has no friction or air drag?

2. What are some sources of energy loss besides friction between two surfaces?

Applications of Conservation of Energy – Lesson 4

Lesson Objectives

• Know how energy and kinematics can find solutions to the same problems

• Apply conservation of energy to roller coasters

• Predict motion including air resistance

Vocabulary

Differential equations

Terminal velocity

Check Your Understanding

1. A ball is held in the air at rest and has 100J of potential energy. Half way down, how much total energy, kinetic energy, and potential energy does it have?

2. A roller coaster car starts off with 10,000J of energy. At the end of the ride, it has 4,000J of energy. What must have happened?

3. Explain the equation: W E= ∆

Introduction

Energy method can be applied to some of the same problems solved by force method and using the equations of motion. In many cases, energy method can allow for solutions to problems that are not possible with either of those methods. The use of the concept of energy allows you to ignore what happens to a system between the two chosen times to compare. Even if energy is lost due to friction or other sources, W E= ∆ still allows you to discover what has happened overall to the state of energy of the system. Energy method even may seem to make problems easier, but it takes practice applying it. That is the goal of this lesson.

Lesson Content

Energy Method versus Kinematics

Imagine a projectile fired from the top of a cliff. The study of kinematics shows how to find the final velocity of the projectile given information about its initial velocity and position. Energy method will do this as well.

Example: A ball is kicked from the top of a building so that it has an initial velocity of 22.0m/s and an angle of 35 degrees. The roof is 3.2m above the flat ground below. Use kinematics and energy method to find the final velocity.

Solution:

The kinematic solution begins with

finding the components of the initial

velocity.

( )0 22.0 cos35 18.0m mx s sv = =

( )0 22.0 sin 35 12.6m my s sv = =

Use the vy2 equation to find final y

velocity:

( ) ( ) ( )2

22 20 2 12.6 19.6 3.2m m

y y s sv v g y m= − ∆ = − −

14.9 my sv = − and 0 18.0 m

x x sv v= =

2 21 23.4 m

x y sv v v= + =

The Energy method selects the 0 and 1 position as the start and finish. Since no

work is done by non-conservative forces, 0 1E E= . Expressions for each of these

total energies can be found and set equal to each other. 21

0 0 0 0 02E PE KE mgh mv= + = + ( ) 211 1 1 120E PE KE mg mv= + = + Since 0 1E E= ,

2 21 10 0 12 2mgh mv mv+ = The masses cancel out: 2

1 0 02 23.4 msv v gh= + = Same result!

Roller Coasters, Pendulums, and other Energy method friendly cases

The results of both methods in the example above are the same. Different methods should yield the same results, or the interpretation by the scientist is wrong or perhaps the theory itself is wrong. That is science. The energy method can do things that kinematics cannot. Kinematics relies on mathematically predictable accelerations. If the path between two points is curvy and unpredictable, kinematics would become increasingly complex and probably become unusable. Energy method only requires knowledge of the start and finish, and then it compares the energies.

Consider the case of a pendulum shown in figure 6, which consists of a mass tied to a string allowed to swing back and forth. For our physics world example, let there be no friction or air resistance.

The diagram on the top shows a pendulum that is released from rest. The burgundy shows the initial position, and the green and blue show the pendulum at later times.

There are two plots sharing the same axes showing potential and kinetic energy. The kinetic energy starts at zero since the pendulum begins with no velocity. It is at its maximum height, so potential energy is at a maximum. As the pendulum is released, it loses potential energy and gains kinetic energy. In chapter 8, it will become clear why this plot takes on the shape of a “sine” curve.

Notice that the total amount of energy at any time is always the same. The units of this plot are normalized so the maximum energy is 1.0. The two plots cross each other at 0.5, so the combined total is still 1.0. This same conversion of energy occurs with any system that involved changing height and velocity in a constant gravity field.

Figure 6. The changing potential and kinetic energy of a pendulum

Online Simulation of a Pendulum and changing energy types

http://demonstrations.wolfram.com/AnimatedPendulum/

Similar to a pendulum is a roller coaster. Most roller coasters use some mechanical means to lift the car to a high position where it has a lot of potential energy. At that point, the roller coaster is released and uses that initial potential energy to drive the roller coaster through the rest of the ride. There is a constant conversion of potential and kinetic energies. In real roller coaster design, there is always some friction but even this can be included using W E= ∆ .

In physics world, roller coasters can be frictionless. Figure 7 shows an idealized case of a roller coaster and how the potential and kinetic energies trade back and forth while keeping the total energy the same. Notice how the roller coaster goes below the reference level and has a negative height and thus a negative potential energy. To compensate, the roller coaster takes on an even larger kinetic energy so the total energy stayed the same.

Figure 7. The ideal frictionless roller coaster

Example – Bad idea for an amusement park ride

An idea for an amusement park ride involves riders sliding down a hill that is frictionless and continuing to slide up to a plateau where a section of carpet will bring them to a halt. Given that hA = 24.0m, hC = 16.0m, and that L = 12.0m, find:

a) The speed of the rider at point B

b) The coefficient of friction of the carpet/rider necessary to stop the rider

Solution:

PART A) Since there is no friction between point A and B, the energies at those points will be

equal. A BE E= 0A A A AE PE KE mgh= + = + and 2120B B B BE PE KE mv= + = +

Setting these two equal to each other gives: 212A Bmgh mv= and 2 21.7 m

B A sv gh= =

PART B) W E= ∆ . Each of these two sides can be found separately and then set equal.

D AE E E∆ = − -- Since the energy at A is the same as C, it is easier to just use the energy at

A.

( )C A A CE mgh mgh mg h h∆ = − = − −

f fW F L= − -- The work done by friction is always negative since the angle between Ff and L is

180 degrees, and cosine (180o) = -1. The force of friction on a flat surface is: f kF mgµ= .

W E= ∆ becomes ( )k A CmgL mg h hµ− = − − and ( ) ( )24.0 16.0

0.6712.0

A Ck

h h m mL m

µ− −

= = =

Energy with Air Resistance and other variable forces

Understanding falling with air resistance will require a re-visit of force method. Air resistance can be explained as a topic of fluid dynamics, where there are terms for drag at high velocity and terms for drag at low velocities. For a detailed view of these terms and how they are used to analyze the effect of air resistance, see the video on Resistive Forces at MIT Open Courseware for Classical Mechanics (Lewin, Fall 1999)

http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-12/

Only cases of low velocity drag will be considered here, where the drag force depends linearly on the velocity. dragF kv= − . This sets up a force problem where a free falling body has an

extra force on it beside weight force. Figure 8 shows the free body diagram for an object with air drag. Applying the force method, step 2 and 3 combine to give the equation for the net forces on the falling object.

F ma= −∑ kv mg ma− = − . Replacing acceleration with its definition as the derivative of

velocity, dividing by mass and switching the minus signs produces:

km

dvg vdt

− = This is an equation that has a function and its derivative together. These are

called differential equations. For simple differential equations, there is a technique to use regular calculus to help find solutions. It requires rearranging the differential equation so that

there is the differential and the variable in the form duu

. Right now, the air drag force equation

needs some sort of substitution to help it take this form. Let kmu g v= − and then k

mdu dv= − .

Solving for dv, multiplying by dt, and dividing by u gives this result:

km

dudtu

− = . Now, integrate both sides. 00

t ukm u

dudtu

− =∫ ∫ 0

lnkm

utu

− =

where 0 0

kmu g v= −

Substituting the u’s and solving for velocity as a function of time offers:

( ) ( )0

k tmg mg mk kv t v e

−= + − . After a long time, the velocity will become mostly constant. This

constant velocity reached after falling for a while is called terminal velocity.

Figure 8. FBD with Air Drag Figure 9. Plot of velocity with air drag

Example of Energy with Air Resistance

As another bad idea for an amusement park ride, a rider is dropped from a height of 1060m into a container of toy prize animals. The rider has a mass of 60kg and reaches 99% of their terminal velocity of 120 54miles m

hour s= .

a) Show that the drag constant is 11 N skgk ⋅= .

b) How much work is done by air resistance during the fall?

Solution:

0

10

20

30

40

50

60

0 20 40 60

Velo

city

(m/s

)

Time (seconds)

Velocity with Air Resistance Fdrag= -kv, m = 60kg, k = 11 Ns/kg

PART A) From the force method developed for air drag, kv mg ma− = − . At

terminal velocity, acceleration is zero. Solving for the constant,

( ) ( )260 9.810.89 11

54

ms N s N s

kg kgms

kgmgkv

⋅ ⋅= = = ≈

PART B) If there were no air resistance, the rider would have the same amount of

energy they had before they were dropped. However, after falling for 1060m, the

rider is within 1% of their terminal velocity (this is left as an exercise to integrate

the velocity equation to confirm this). The solution begins with:

( ) ( ) ( )( ) ( ) ( ) ( )2

221 12 20.99 60 0.99 54 60 9.8 1060m m

drag final original t s sW E E E m v mgh kg kg m= ∆ = − = − = −

540000dragW J= − . 86% of the energy is lost due to air resistance for this rider to

nearly reach terminal velocity.

Lesson Summary

• Conservation of Energy can be used to solve some of the same problems as kinematics and force method.

• Energy method only requires information about a start and chosen finish time and is path independent like work calculations.

• Pendula, and roller coasters are examples of scenarios where kinematics and forces may not provide a solution, but energy can.

• W E= ∆ is an ideal starting place for energy method. Work and the change in energy can be found separately and then put back in the relationship to solve for unknowns.

Review Questions

Review Problems

<include problem with a variable force that requires an integral to find work in an energy problem>

Further Reading / Supplemental Links

Points to Consider

1. How would the force method be useful for structures that do not move, like a bridge?

2. Can the sum of forces equal to zero for an object that is moving?