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# Chapter 2. Conservation of Momentum

Fantasy novelist T.H. White invented a wonderful phrase that has since entered into popular culture: “Everything not forbidden is compulsory.” Originally intended as a satire of totalitarianism, it was taken up by physicist Murray Gell-Mann as a metaphor for physics. What he meant was that the laws of physics forbid all the impossible things, and what's left over is what really happens. Conservation of mass and energy prevent many things from happening. Objects can't disappear into thin air, and you can't run your car forever without putting gas in it.

Some other processes are impossible, but not forbidden by these two conservation laws. In the martial arts movie Crouching Tiger, Hidden Dragon, those who have received mystical enlightenment are able to violate the laws of physics. Some of the violation, such as their ability to fly, are obvious, but others are a little more subtle. The rebellious young heroine/antiheroine Jen Yu gets into an argument while sitting at a table in a restaurant. A young tough, Iron Arm Lu, comes running toward her at full speed, and she puts up one arm and effortlessly makes him bounce back, without even getting out of her seat or bracing herself against anything. She does all this between bites. It's impossible, but how do we know it's impossible? It doesn't violate conservation of mass, because neither character's mass changes. It conserves energy as well, since the rebounding Lu has the same energy he started with.

Suppose you live in a country where the only laws are prohibitions against murder and robbery. One day someone covers your house with graffiti, and the authorities refuse to prosecute, because no crime was committed. You're convinced of the need for a new law against vandalism. Similarly, the story of Jen Yu and Iron Arm Lu shows that we need a new conservation law.

## 2.1 Translation Symmetry

The most fundamental laws of physics are conservation laws, and Noether's theorem tells us that conservation laws are the way they are because of symmetry. Time symmetry is responsible for conservation of energy, but time is like a river with only two directions, past and future. What's impossible about Lu's motion is the abrupt reversal in the direction of his motion in space, but neither time symmetry nor energy conservation tell us anything about directions in space. When you put gas in your car, you don't have to decide whether you want to buy north gas or south gas, east, west, up or down gas. Energy has no direction. What we need is a new conserved quantity that has a direction in space, and such a conservation law can only come from a symmetry that relates to space. Since we've already had some luck with time symmetry, which says that the laws of physics are the same at all times, it seems reasonable to turn now to the possibility of a new type of symmetry, which would state that the laws of physics are the same in all places in space. This is known as translation symmetry, where the word “translation” is being used in a mathematical sense that means sliding something around without rotating it.

Translation symmetry would seem reasonable to most people, but you'll see that it ends up producing some very surprising results. To see how, it will be helpful to imagine the consequences of a violation of translation symmetry. What if, like the laws of nations, the laws of physics were different in different places? What would happen, and how would we detect it? We could try doing the same experiment in two different places and comparing the results, but it's even easier than that. Tap your finger on this spot on the page

$\begin{equation*} \times \end{equation*}$

and then wait a second and do it again. Did both taps occur at the same point in space? You're probably thinking that's a silly question; am I just checking whether you followed my directions? Not at all. Consider the whole scene from the point of view of a Martian who is observing it through a powerful telescope from her home planet. (You didn't draw the curtains, did you?) From her point of view, the earth is spinning on its axis and orbiting the sun, at speeds measured in thousands of kilometers per hour. According to her, your second finger tap happened at a point in space about 30 kilometers from the first. If you want to impress the Martians and win the Martian version of the Nobel Prize for detecting a violation of translation symmetry, all you have to do is perform a physics experiment twice in the same laboratory, and show that the result is different.

But who's to say that the Martian point of view is the right one? It gets a little thorny now. How do you know that what you detected was a violation of translation symmetry at all? Maybe it was just a violation of time symmetry. The Martian Nobel committee isn't going to give you the prize based on an experiment this ambiguous. A possible scheme for resolving the ambiguity would be to wait a year and do the same experiment a third time. After a year, the earth will have completed one full orbit around the sun, and your lab will be back in the same spot in space. If the third experiment comes out the same as the first one, then you can make a strong argument that what you've detected is an asymmetry of space, not time. There's a problem, however. You and the Martians agree that the earth is back in the same place after a year, but what about an observer from another solar system, whose planet orbits a different star? This observer says that our whole solar system is in motion. To him, the earth's motion around our sun looks like a spiral or a corkscrew, since the sun is itself moving.

## 2.2 The Principle of Inertia

### Symmetry and inertia

This story shows that translation symmetry is closely related to the relative nature of motion, as expressed by the principle of inertia. Riding in a train on a long, straight track at constant speed, how can you even tell you're in motion? You can look at the scenery outside, but that's irrelevant, because we could argue that the trees and cows are moving while you stand still. (The Martians say both train and scenery are moving.) The real point is whether you can detect your motion without reference to any external object. You can hear the repetitive thunk-thunk-thunk as the train passes from one piece of track to the next, but again this is just a reference to an external object --- all that proves is that you're moving relative to the tracks, but is there any way to tell that you're moving in some absolute sense? Assuming no interaction with the outside world, is there any experiment you can do that will give a different result when the train is in motion than when it's at rest? You could if translation symmetry was violated. If the laws of physics were different in different places, then as the train moved it would pass through them. “Riding over” these regions would be like riding over the pieces of track, but you would be able to detect the transition from one region to the next simply because experiments inside the train came out different, without referring to any external objects. Rather than the thunk-thunk-thunk of the rails, you would detect increases and decreases in some quantity such as the gravitational constant $$G$$, or the speed of light, or the mass of the electron.

We can therefore conclude that the following two hypotheses are closely related.

##### The principle of inertia

The results of experiments don't depend on the straight-line, constant-speed motion of the apparatus.

##### Translation symmetry

The laws of physics are the same at every point in space. Specifically, experiments don't give different results just because you set up your apparatus in a different place.

##### Example 1: A state of absolute rest
Suppose that translation symmetry is violated. The laws of physics are different in one region of space than in another. Cruising in our spaceship, we monitor the fluctuations in the laws of physics by watching the needle on a meter that measures some fundamental quantity such as the gravitational constant. We make a short blast with the ship's engines and turn them off again. Now we see that the needle is wavering more slowly, so evidently it's taking us more time to move from one region to the next. We keep on blasting with the ship's engines until the fluctuations stop entirely. Now we know that we're in a state of absolute rest. The violation of translation symmetry logically resulted in a violation of the principle of inertia.
self-check:

Suppose you do an experiment to see how long it takes for a rock to drop one meter. This experiment comes out different if you do it on the moon. Does this violate translation symmetry?

## 2.3 Momentum

d / A less cumbersome representation of figure c. The collision is shown as a graph of position, $$x$$, versus time, $$t$$. By distorting the graph-paper grid, we can describe the same collision in the two other frames of reference. Cf. figure e.

e / A visual shorthand for describing the relationship between two frames of reference as in figure d: the gray rectangle represents the original, undistorted, graph paper, while the superimposed grid shows a different frame.

f / Example 3.

g / Example 5.

h / Example 8.

i / It doesn't make sense to add his debts to her assets.

j / I squeeze the bathroom scale. It does make sense to add my fingers' force to my thumbs', because they both act on the same object --- the scale.

k / A ball is falling (or rising).

l / The same ball is viewed in a frame of reference that is moving horizontally.

m / The drops of water travel in parabolic arcs.

n / Example 12.

o / The memory of motion: the default would be for the ball to continue doing what it was already doing. The force of gravity makes it deviate downward, ending up one square below the default.

p / The forces on car 1 cancel, and the total force on it is zero. The forward and backward forces on car 2 also cancel. Only the inward force remains.

### Conservation of momentum

Let's return to the impossible story of Jen Yu and Iron Arm Lu on page 37. For simplicity, we'll model them as two identical, featureless pool balls, a. This may seem like a drastic simplification, but even a collision between two human bodies is really just a series of many collisions between atoms. The film shows a series of instants in time, viewed from overhead. The light-colored ball comes in, hits the darker ball, and rebounds. It seems strange that the dark ball has such a big effect on the light ball without experiencing any consequences itself, but how can we show that this is really impossible?

a / How can we prove that this collision is impossible?

We can show it's impossible by looking at it in a different frame of reference, b. This camera follows the light ball on its way in, so in this frame the incoming light ball appears motionless. (If you ever get hauled into court on an assault charge for hitting someone, try this defense: “Your honor, in my fist's frame of reference, it was his face that assaulted my knuckles!”) After the collision, we let the camera keep moving in the same direction, because if we didn't, it wouldn't be showing us an inertial frame of reference. To help convince yourself that figures a and b represent the same motion seen in two different frames, note that both films agree on the distances between the balls at each instant. After the collision, frame b shows the light ball moving twice as fast as the dark ball; an observer who prefers frame a explains this by saying that the camera that produced film b was moving one way, while the ball was moving the opposite way.

b / The collision of figure a is viewed in a different frame of reference.

Figures a and b record the same events, so if one is impossible, the other is too. But figure b is definitely impossible, because it violates conservation of energy. Before the collision, the only kinetic energy is the dark ball's. After the collision, light ball suddenly has some energy, but where did that energy come from? It can only have come from the dark ball. The dark ball should then have lost some energy, which it hasn't, since it's moving at the same speed as before.

Figure c shows what really does happen. This kind of behavior is familiar to anyone who plays pool. In a head-on collision, the incoming ball stops dead, and the target ball takes all its energy and flies away. In c/1, the light ball hits the dark ball. In c/2, the camera is initially following the light ball; in this frame of reference, the dark ball hits the light one (“Judge, his face hit my knuckles!”). The frame of reference shown in c/3 is particularly interesting. Here the camera always stays at the midpoint between the two balls. This is called the center-of-mass frame of reference.

c / This is what really happens. Three films represent the same collision viewed in three different frames of reference. Energy is conserved in all three frames. Figure d shows a less cumbersome way of representing the same thing.

self-check:

In each picture in figure c/1, mark an x at the point half-way in between the two balls. This series of five x's represents the motion of the camera that was used to make the bottom film. How fast is the camera moving? Does it represent an inertial frame of reference?

What's special about the center-of-mass frame is its symmetry. In this frame, both balls have the same initial speed. Since they start out with the same speed, and they have the same mass, there's no reason for them to behave differently from each other after the collision. By symmetry, if the light ball feels a certain effect from the dark ball, the dark ball must feel the same effect from the light ball.

This is exactly like the rules of accounting. Let's say two big corporations are doing business with each other. If Glutcorp pays a million dollars to Slushco, two things happen: Glutcorp's bank account goes down by a million dollars, and Slushco's rises by the same amount. The two companies' books have to show transactions on the same date that are equal in size, but one is positive (a payment) and one is negative. What if Glutcorp records $$-1,000,000$$ dollars, but Slushco's books say $$+920,000$$? This indicates that a law has been broken; the accountants are going to call the police and start looking for the employee who's driving a new 80,000-dollar Jaguar. Money is supposed to be conserved.

In figure c, let's define velocities as positive if the motion is toward the top of the page. In figure c/1 let's say the incoming light ball's velocity is 1 m/s.

 before the collision after the collision small ball 0 1 big ball + 1 1

The books balance. The light ball's payment, $$-1$$, matches the dark ball's receipt, $$+1$$. Everything also works out fine in the center of mass frame, c/3:

 before the collision after the collision small ball ensuremath05 ensuremath05 big ball + 1 ensuremath05
self-check:

Make a similar table for figure c/2. What do you notice about the change in velocity when you compare the three tables?

Accounting works because money is conserved. Apparently, something is also conserved when the balls collide. We call it momentum. Momentum is not the same as velocity, because conserved quantities have to be additive. Our pool balls are like identical atoms, but atoms can be stuck together to form molecules, people, and planets. Because conservation laws work by addition, two atoms stuck together and moving at a certain velocity must have double the momentum that a single atom would have had. We therefore define momentum as velocity multiplied by mass.

##### Conservation of momentum

The quantity defined by $$\text{momentum} = mv$$ is conserved.

This is our second example of Noether's theorem:\nopagebreak

\begin{noethertable} time symmetry & \noetherimplies & mass-energy
translation symmetry & \noetherimplies & momentum
\end{noethertable}

##### Example 2: Conservation of momentum for pool balls

$$\triangleright$$ Is momentum conserved in figure c/1?

$$\triangleright$$ We have to check whether the total initial momentum is the same as the total final momentum.

$\begin{multline*} \text{dark ball's initial momentum} + \text{light ball's initial momentum}\\ =? \\ \text{dark ball's final momentum} + \text{light ball's final momentum} \end{multline*}$

Yes, momentum was conserved:

$\begin{equation*} 0+mv = mv+0 \end{equation*}$

##### Example 3: Ice skaters push off from each other
If the ice skaters in figure f have equal masses, then left-right (mirror) symmetry implies that they moved off with equal speeds in opposite directions. Let's check that this is consistent with conservation of momentum:
$\begin{multline*} \text{left skater's initial momentum} + \text{right skater's initial momentum}\\ =? \\ \text{left skater's final momentum} + \text{right skater's final momentum} \end{multline*}$
Momentum was conserved:
$\begin{equation*} 0+0 = m\times(-v)+mv \end{equation*}$
This is an interesting example, because if these had been pool balls instead of people, we would have accused them of violating conservation of energy. Initially there was zero kinetic energy, and at the end there wasn't zero. (Note that the energies at the end don't cancel, because kinetic energy is always positive, regardless of direction.) The mystery is resolved because they're people, not pool balls. They both ate food, and they therefore have chemical energy inside their bodies:
$\begin{equation*} \text{food energy} \rightarrow \text{kinetic energy} + \text{kinetic energy} + \text{heat} \end{equation*}$
##### Example 4: Unequal masses

$$\triangleright$$ Suppose the skaters have unequal masses: 50 kg for the one on the left, and 55 kg for the other. The more massive skater, on the right, moves off at 1.0 m/s. How fast does the less massive skater go?

$$\triangleright$$ Their momenta (plural of momentum) have to be the same amount, but with opposite signs. The less massive skater must have a greater velocity if her momentum is going to be as much as the more massive one's.

\begin{align*} 0+0 &= (50\ \text{kg})(-v)+(55\ \text{kg})(1.0\ \text{m}/\text{s})\\ (50\ \text{kg})(v) &= (55\ \text{kg})(1.0\ \text{m}/\text{s})\\ v &= \frac{(55\ \text{kg})}{50\ \text{kg}}(1.0\ \text{m}/\text{s})\\ &= 1.1\ \text{m}/\text{s} \end{align*}

### Momentum compared to kinetic energy

Momentum and kinetic energy are both measures of the amount of motion, and a sideshow in the Newton-Leibniz controversy over who invented calculus was an argument over which quantity was the “true” measure of motion. The modern student can certainly be excused for wondering why we need both quantities, when their complementary nature was not evident to the greatest minds of the 1700's. The following table highlights their differences.

 Kinetic energy… Momentum… has no direction in space. has a direction in space. is always positive, and cannot cancel out. cancels with momentum in the opposite direction. can be traded for forms of energy that do not involve motion. KE is not a conserved quantity by itself. is always conserved. is quadrupled if the velocity is doubled. is doubled if the velocity is doubled.

Here are some examples that show the different behaviors of the two quantities.

##### Example 5: A spinning coin
A spinning coin has zero total momentum, because for every moving point, there is another point on the opposite side that cancels its momentum. It does, however, have kinetic energy.
##### Example 6: Momentum and kinetic energy in firing a rifle

The rifle and bullet have zero momentum and zero kinetic energy to start with. When the trigger is pulled, the bullet gains some momentum in the forward direction, but this is canceled by the rifle's backward momentum, so the total momentum is still zero. The kinetic energies of the gun and bullet are both positive numbers, however, and do not cancel. The total kinetic energy is allowed to increase, because both objects' kinetic energies are destined to be dissipated as heat --- the gun's “backward” kinetic energy does not refrigerate the shooter's shoulder!

##### Example 7: The wobbly earth

As the moon completes half a circle around the earth, its motion reverses direction. This does not involve any change in kinetic energy, because the moon doesn't speed up or slow down, nor is there any change in gravitational energy, because the moon stays at the same distance from the earth.1 The reversed velocity does, however, imply a reversed momentum, so conservation of momentum tells us that the earth must also change its momentum. In fact, the earth wobbles in a little “orbit” about a point below its surface on the line connecting it and the moon. The two bodies' momenta always point in opposite directions and cancel each other out.

##### Example 8: The earth and moon get a divorce
Why can't the moon suddenly decide to fly off one way and the earth the other way? It is not forbidden by conservation of momentum, because the moon's newly acquired momentum in one direction could be canceled out by the change in the momentum of the earth, supposing the earth headed the opposite direction at the appropriate, slower speed. The catastrophe is forbidden by conservation of energy, because both their kinetic energies would have increased greatly.
##### Example 9: Momentum and kinetic energy of a glacier

A cubic-kilometer glacier would have a mass of about $$10^{12}$$ kg --- 1 followed by 12 zeroes. If it moves at a speed of $$0.00001$$ m/s, then its momentum would be $$10,000,000\ \text{kg}\!\cdot\!\text{m}/\text{s}$$. This is the kind of heroic-scale result we expect, perhaps the equivalent of the space shuttle taking off, or all the cars in LA driving in the same direction at freeway speed. Its kinetic energy, however, is only 50 joules, the equivalent of the calories contained in a poppy seed or the energy in a drop of gasoline too small to be seen without a microscope. The surprisingly small kinetic energy is because kinetic energy is proportional to the square of the velocity, and the square of a small number is an even smaller number.

### Force

#### Definition of force

When momentum is being transferred, we refer to the rate of transfer as the force.3 The metric unit of force is the newton (N). The relationship between force and momentum is like the relationship between power and energy, or the one between your cash flow and your bank balance:

 conserved quantity rate of transfer name units name units energy joules (J) power watts (W) momentum textupkgcdottextupmtextups force newtons (N)
##### Example 10: A bullet
$$\triangleright$$ A bullet emerges from a gun with a momentum of 1.0 $$\text{kg}\!\cdot\!\text{m}/\text{s}$$, after having been acted on for 0.01 seconds by the force of the gases from the explosion of the gunpowder. What was the force on the bullet?

$$\triangleright$$ The force is2

$\begin{equation*} \frac{1.0}{0.01} = 100\ \text{newtons} . \end{equation*}$

There's no new physics happening here, just a definition of the word “force.” Definitions are neither right nor wrong, and just because the Chinese call it \raisebox{-0.2mm}{} instead, that doesn't mean they're incorrect. But when Isaac Newton first started using the term “force” according to this technical definition, people already had some definite ideas about what the word meant.

In some cases Newton's definition matches our intuition. In example 10, we divided by a small time, and the result was a big force; this is intuitively reasonable, since we expect the force on the bullet to be strong.

#### Forces occur in equal-strength pairs

In other situations, however, our intuition rebels against reality.

##### Example 11: Extra protein
$$\triangleright$$ While riding my bike fast down a steep hill, I pass through a cloud of gnats, and one of them goes into my mouth. Compare my force on the gnat to the gnat's force on me.

$$\triangleright$$ Momentum is conserved, so the momentum gained by the gnat equals the momentum lost by me. Momentum conservation holds true at every instant over the fraction of a second that it takes for the collision to happen. The rate of transfer of momentum out of me must equal the rate of transfer into the gnat. Our forces on each other have the same strength, but they're in opposite directions.

Most people would be willing to believe that the momentum gained by the gnat is the same as the momentum lost by me, but they would not believe that the forces are the same strength. Nevertheless, the second statement follows from the first merely as a matter of definition. Whenever two objects, A and B, interact, A's force on B is the same strength as B's force on A, and the forces are in opposite directions.4

$\begin{equation*} \text{(A on B)} = -\text{(B on A)} \end{equation*}$

Using the metaphor of money, suppose Alice and Bob are adrift in a life raft, and pass the time by playing poker. Money is conserved, so if they count all the money in the boat every night, they should always come up with the same total. A completely equivalent statement is that their cash flows are equal and opposite. If Alice is winning five dollars per hour, then Bob must be losing at the same rate.

This statement about equal forces in opposite directions implies to many students a kind of mystical principle of equilibrium that explains why things don't move. That would be a useless principle, since it would be violated every time something moved.5 The ice skaters of figure f on page 44 make forces on each other, and their forces are equal in strength and opposite in direction. That doesn't mean they won't move. They'll both move --- in opposite directions.

The fallacy comes from trying to add things that it doesn't make sense to add, as suggested by the cartoon in figure i. We only add forces that are acting on the same object. It doesn't make sense to say that the skaters' forces on each other add up to zero, because it doesn't make sense to add them. One is a force on the left-hand skater, and the other is a force on the right-hand skater.

In figure j, my fingers' force and my thumbs' force are both acting on the bathroom scale. It does make sense to add these forces, and they may possibly add up to zero, but that's not guaranteed by the laws of physics. If I throw the scale at you, my thumbs' force is stronger that my fingers', and the forces no longer cancel:

$\begin{equation*} \text{(fingers on scale)} \ne -\text{(thumbs on scale)} . \end{equation*}$

What's guaranteed by conservation of momentum is a whole different relationship:

\begin{align*} \text{(fingers on scale)} &= -\text{(scale on fingers)} \\ \text{(thumbs on scale)} &= -\text{(scale on thumbs)} \\ \end{align*}

#### The force of gravity

How much force does gravity make on an object? From everyday experience, we know that this force is proportional to the object's mass.6 Let's find the force on a one-kilogram object. If we release this object from rest, then after it has fallen one meter, its kinetic energy equals the strength of the gravitational field,

$\begin{equation*} 10\ \text{joules per kilogram per meter}\times1\ \text{kilogram}\times1\ \text{meter} = 10\ \text{joules} . \end{equation*}$

Using the equation for kinetic energy and doing a little simple algebra, we find that its final velocity is 4.4 m/s. It starts from 0 m/s, and ends at 4.4 m/s, so its average velocity is 2.2 m/s, and the time takes to fall one meter is therefore (1 m)/(2.2 m/s)=0.44 seconds. Its final momentum is 4.4 units, so the force on it was evidently

$\begin{equation*} \frac{4.4}{0.44} = 10\ \text{newtons} . \end{equation*}$

This is like one of those card tricks where the magician makes you go through a bunch of steps so that you end up revealing the card you had chosen --- the result is just equal to the gravitational field, 10, but in units of newtons! If algebra makes you feel warm and fuzzy, you may want to replay the derivation using symbols and convince yourself that it had to come out that way. If not, then I hope the numerical result is enough to convince you of the general fact that the force of gravity on a one-kilogram mass equals $$g$$. For masses other than one kilogram, we have the handy-dandy result that

$\begin{equation*} (\text{force of gravity on a mass m}) = mg . \end{equation*}$

In other words, $$g$$ can be interpreted not just as the gravitational energy per kilogram per meter of height, but also as the gravitational force per kilogram.

### Motion in two dimensions

#### Projectile motion

Galileo was an innovator in more than one way. He was arguably the inventor of open-source software: he invented a mechanical calculating device for certain engineering applications, and rather than keeping the device's design secret as his competitors did, he made it public, but charged students for lessons in how to use it. Not only that, but he was the first physicist to make money as a military consultant. Galileo understood projectiles better than anyone else, because he understood the principle of inertia. Even if you're not planning on a career involving artillery, projectile motion is a good thing to learn about because it's an example of how to handle motion in two or three dimensions.

Figure k shows a ball in the process of falling --- or rising, it really doesn't matter which. Let's say the ball has a mass of one kilogram, each square in the grid is 10 meters on a side, and the positions of the ball are shown at time intervals of one second. The earth's gravitational force on the ball is 10 newtons, so with each second, the ball's momentum increases by 10 units, and its speed also increases by 10 m/s. The ball falls 10 m in the first second, 20 m in the next second, and so on.

self-check:

What would happen if the ball's mass was 2 kilograms?

Now let's look at the ball's motion in a new frame of reference, l, which is moving at 10 meters per second to the left compared to the frame of reference used in figure k. An observer in this frame of reference sees the ball as moving to the right by 10 meters every second. The ball traces an arc of a specific mathematical type called a parabola:

\parbox{80mm}{

 1 step over and 1 step down 1 step over and 2 steps down 1 step over and 3 steps down 1 step over and 4 steps down …

}

It doesn't matter which frame of reference is the “real” one. Both diagrams show the possible motion of a projectile. The interesting point here is that the vertical force of gravity has no effect on the horizontal motion, and the horizontal motion also has no effect on what happens in the vertical motion. The two are completely independent. If the sun is directly overhead, the motion of the ball's shadow on the ground seems perfectly natural: there are no horizontal forces, so it either sits still or moves at constant velocity. (Zero force means zero rate of transfer of momentum.) The same is true if we shine a light from one side and cast the ball's shadow on the wall. Both shadows obey the laws of physics.

##### Example 12: The moon
In example 12 on page 27, I promised an explanation of how Newton knew that the gravitational field experienced by the moon due to the earth was 1/3600 of the one we feel here on the earth's surface. The radius of the moon's orbit had been known since ancient times, so Newton knew its speed to be 1,100 m/s (expressed in modern units). If the earth's gravity wasn't acting on the moon, the moon would fly off straight, along the straight line shown in figure n, and it would cover 1,100 meters in one second. We observe instead that it travels the arc of a circle centered on the earth. Straightforward geometry shows that the amount by which the arc drops below the straight line is 1.6 millimeters. Near the surface of the earth, an object falls 5 meters in one second,7 which is indeed about 3600 times greater than 1.6 millimeters.

The tricky part about this argument is that although I said the path of a projectile was a parabola, in this example it's a circle. What's going on here? What's different here is that as the moon moves 1,100 meters, it changes its position relative to the earth, so down is now in a new direction. We'll discuss circular motion more carefully soon, but in this example, it really doesn't matter. The curvature of the arc is so gentle that a parabola and a circle would appear almost identical. (Actually the curvature is so gentle --- 1.6 millimeters over a distance of 1,100 meters! --- that if I had drawn the figure to scale, you wouldn't have even been able to tell that it wasn't straight.)

As an interesting historical note, Newton claimed that he first did this calculation while confined to his family's farm during the plague of 1666, and found the results to “answer pretty nearly.” His notebooks, however, show that although he did the calculation on that date, the result didn't quite come out quite right, and he became uncertain about whether his theory of gravity was correct as it stood or needed to be modified. Not until 1675 did he learn of more accurate astronomical data, which convinced him that his theory didn't need to be tinkered with. It appears that he rewrote his own life story a little bit in order to make it appear that his work was more advanced at an earlier date, which would have helped him in his dispute with Leibniz over priority in the invention of calculus.

#### The memory of motion

There's another useful way of thinking about motion along a curve. In the absence of a force, an object will continue moving in the same speed and in the same direction. One of my students invented a wonderful phrase for this: the memory of motion. Over the first second of its motion, the ball in figure o moved 1 square over and 1 square down, which is 10 meters and 10 meters. The default for the next one-second interval would be to repeat this, ending up at the location marked with the first dashed circle. The earth's 10-newton gravitational force on the ball, however, changes the vertical part of the ball's momentum by 10 units. The ball actually ends up 10 meters (1 square) below the default.

#### Circular motion

Figure q shows how to apply the memory-of-motion idea to circular motion. It should convince you that only an inward force is needed to produce circular motion. One of the reasons Newton was the first to make any progress in analyzing the motion of the planets around the sun was that his contemporaries were confused on this point. Most of them thought that in addition to an attraction from the sun, a second, forward force must exist on the planets, to keep them from slowing down. This is incorrect Aristotelian thinking; objects don't naturally slow down. Car 1 in figure p only needs a forward force in order to cancel out the backward force of friction; the total force on it is zero. Similarly, the forward and backward forces on car 2 are canceling out, and the only force left over is the inward one. There's no friction in the vacuum of outer space, so if car 2 was a planet, the backward force wouldn't exist; the forward force wouldn't exist either, because the only force would be the force of the sun's gravity.

q / A large number of gentle taps gives a good approximation to circular motion. A steady inward force would give exactly circular motion.

One confusing thing about circular motion is that it often tempts us psychologically to adopt a noninertial frame of reference. Figure r shows a bowling ball in the back of a turning pickup truck. Each panel gives a view of the same events from a different frame of reference. The frame of reference r/1, attached to the turning truck, is noninertial, because it changes the direction of its motion. The ball violates conservation of energy by accelerating from rest for no apparent reason. Is there some mysterious outward force that is slamming the ball into the side of the truck's bed? No. By analyzing everything in a proper inertial frame of reference, r/2, we see that it's the truck that swerves and hits the ball. That makes sense, because the truck is interacting with the asphalt.

r / A bowling ball is in the back of a pickup truck turning left. The motion is viewed first in a frame that turns along with the truck, 1, and then in an inertial frame, 2.

## 2.4 Newton's Triumph

t / Tycho Brahe made his name as an astronomer by showing that the bright new star, today called a supernova, that appeared in the skies in 1572 was far beyond the Earth's atmosphere. This, along with Galileo's discovery of sunspots, showed that contrary to Aristotle, the heavens were not perfect and unchanging. Brahe's fame as an astronomer brought him patronage from King Frederick II, allowing him to carry out his historic high-precision measurements of the planets' motions. A contradictory character, Brahe enjoyed lecturing other nobles about the evils of dueling, but had lost his own nose in a youthful duel and had it replaced with a prosthesis made of an alloy of gold and silver. Willing to endure scandal in order to marry a peasant, he nevertheless used the feudal powers given to him by the king to impose harsh forced labor on the inhabitants of his parishes. The result of their work, an Italian-style palace with an observatory on top, surely ranks as one of the most luxurious science labs ever built. He died of a ruptured bladder after falling from a wagon on the way home from a party --- in those days, it was considered rude to leave the dinner table to relieve oneself.

u / An ellipse is a circle that has been distorted by shrinking and stretching along perpendicular axes.

v / An ellipse can be constructed by tying a string to two pins and drawing like this with the pencil stretching the string taut. Each pin constitutes one focus of the ellipse.

w / If the time interval taken by the planet to move from P to Q is equal to the time interval from R to S, then according to Kepler's equal-area law, the two shaded areas are equal. The planet is moving faster during interval RS than it did during PQ, which Newton later determined was due to the sun's gravitational force accelerating it. The equal-area law predicts exactly how much it will speed up.

Isaac Newton's greatest triumph was his explanation of the motion of the planets in terms of universal physical laws. It was a tremendous psychological revolution: for the first time, both heaven and earth were seen as operating automatically according to the same rules.

Newton wouldn't have been able to figure out why the planets move the way they do if it hadn't been for the astronomer Tycho Brahe (1546-1601) and his protege Johannes Kepler (1571-1630), who together came up with the first simple and accurate description of how the planets actually do move. The difficulty of their task is suggested by figure s, which shows how the relatively simple orbital motions of the earth and Mars combine so that as seen from earth Mars appears to be staggering in loops like a drunken sailor.

s / As the Earth and Mars revolve around the sun at different rates, the combined effect of their motions makes Mars appear to trace a strange, looped path across the background of the distant stars.

Brahe, the last of the great naked-eye astronomers, collected extensive data on the motions of the planets over a period of many years, taking the giant step from the previous observations' accuracy of about 10 minutes of arc (10/60 of a degree) to an unprecedented 1 minute. The quality of his work is all the more remarkable considering that his observatory consisted of four giant brass protractors mounted upright in his castle in Denmark. Four different observers would simultaneously measure the position of a planet in order to check for mistakes and reduce random errors.

With Brahe's death, it fell to his former assistant Kepler to try to make some sense out of the volumes of data. Kepler, in contradiction to his late boss, had formed a prejudice, a correct one as it turned out, in favor of the theory that the earth and planets revolved around the sun, rather than the earth staying fixed and everything rotating about it. Although motion is relative, it is not just a matter of opinion what circles what. The earth's rotation and revolution about the sun make it a noninertial reference frame, which causes detectable violations of Newton's laws when one attempts to describe sufficiently precise experiments in the earth-fixed frame. Although such direct experiments were not carried out until the 19th century, what convinced everyone of the sun-centered system in the 17th century was that Kepler was able to come up with a surprisingly simple set of mathematical and geometrical rules for describing the planets' motion using the sun-centered assumption. After 900 pages of calculations and many false starts and dead-end ideas, Kepler finally synthesized the data into the following three laws:

##### Kepler's elliptical orbit law

The planets orbit the sun in elliptical orbits with the sun at one focus.

##### Kepler's equal-area law

The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

##### Kepler's law of periods

Let $$T$$, called the planet's period, be the time required for a planet to orbit the sun, and let $$a$$ be the long axis of the ellipse. Then $$T^2$$ is proportional to $$a^3$$.

Although the planets' orbits are ellipses rather than circles, most are very close to being circular. The earth's orbit, for instance, is only flattened by 1.7% relative to a circle. In the special case of a planet in a circular orbit, the two foci (plural of “focus”) coincide at the center of the circle, and Kepler's elliptical orbit law thus says that the circle is centered on the sun. The equal-area law implies that a planet in a circular orbit moves around the sun with constant speed. For a circular orbit, the law of periods then amounts to a statement that $$T^2$$ is proportional to $$r^3$$, where $$r$$ is the radius. If all the planets were moving in their orbits at the same speed, then the time for one orbit would only increase with the circumference of the circle, so we would have a simple proportionality between $$T$$ and $$r$$. Since this is not the case, we can interpret the law of periods to mean that different planets orbit the sun at different speeds. In fact, the outer planets move more slowly than the inner ones.

##### Example 13: Jupiter and Uranus
$$\triangleright$$ The planets Jupiter and Uranus have very nearly circular orbits, and the radius of Uranus's orbit is about four times grater than that of Jupiter's orbit. Compare their orbital periods.

$$\triangleright$$ If all the planets moved at the same speed, then it would take Uranus four times longer to complete the four-times-greater circumference of its orbit. However, the law of periods tells us that this isn't the case. We expect Uranus to take more than four times as long to orbit the sun.

The law of periods is stated as a proportionality, and proportionalities are statements about quantities in proportion to one another, i.e.. about division. We're given information about Uranus's orbital radius divided by Jupiter's, and what we should expect to get out is information about Uranus's period divided by Jupiter's. Let's call the latter ratio $$y$$. Then we're looking for a number $$y$$ such that

\begin{align*} y^2 &= 4^3 , \\ \text{i.e.,} y \times y &= 4 \times 4 \times 4 \\ y \times y &= 64 \\ y &= 8 \end{align*}

The law of periods predicts that Uranus's period will be eight times greater than Jupiter's, which is indeed what is observed (to within the precision to be expected since the given figure of 4 was just stated roughly as a whole number, for convenience in calculation).

What Newton discovered was the reasons why Kepler's laws were true: he showed that they followed from his laws of motion. From a modern point of view, conservation laws are more fundamental than Newton's laws, so rather than following Newton's approach, it makes more sense to look for the reasons why Kepler's laws follow from conservation laws. The equal-area law is most easily understood as a consequence of conservation of angular momentum, which is a new conserved quantity to be discussed in chapter 3. The proof of the elliptical orbit law is a little too mathematical to be appropriate for this book, but the interested reader can find the proof in chapter 15 of my online book Light and Matter.

x / Connecting Kepler's law of periods to the laws of physics.

The law of periods follows directly from the physics we've already covered. Consider the example of Jupiter and Uranus. We want to show that the result of example 13 is the only one that's consistent with conservation of energy and momentum, and Newton's law of gravity. Since Uranus takes eight times longer to cover four times the distance, it's evidently moving at half Jupiter's speed. In figure x, the distance Jupiter covers from A to B is therefore twice the distance Uranus covers, over the same time, from D to E. If there hadn't been any gravitational force from the sun, Jupiter would have ended up at C, and Uranus at F. The distance from B to C is a measure of how much force acted on Jupiter, and likewise for the very small distance from E to F. We find that BC is 16 millimeters on this scale drawing, and EF is 1 mm, but this is exactly what we expect from Newton's law of gravity: quadrupling the distance should give 1/16 the force.

## 2.5 Work

y / The black box does work by reeling in its cable.

Imagine a black box8, containing a gasoline-powered engine, which is designed to reel in a steel cable of length $$d$$, exerting a certain force $$F$$.

If we use this black box was to lift a weight, then by the time it has pulled in its whole cable, it will have lifted the weight through a height $$d$$. The force $$F$$ is barely capable of lifting a weight $$m$$ if $$F=mg$$, and if it does this, then the upward force from the cable exactly cancels the downward force of gravity, so the weight will rise at constant speed, without changing its kinetic energy. Only gravitational energy is transferred into the weight, and the amount of gravitational energy is $$mgd$$, which equals $$Fd$$. By conservation of energy, this must also be the amount of energy lost from the chemical energy of the gasoline inside the box.9

Now what if we use the black box to pull a plow? The energy increase in the outside world is of a different type than before: mainly heat created by friction between the dirt and the ploughshare. The box, however, only communicates with the outside world via the hole through which its cable passes. The amount of chemical energy lost by the gasoline can therefore only depend on $$F$$ and $$d$$, so again the amount of energy transferred must equal $$Fd$$.

The same reasoning can in fact be applied no matter what the cable is being used to do. There must always be a transfer of energy from the box to the outside world that is equal to $$Fd$$. In general, when energy is transferred, we refer to the amount of energy transferred as work, $$W$$. If, as in the example of the black box, the motion of the object to which the force is applied is in the same direction as the force, then $$W=Fd$$.

z / The baseball pitcher put kinetic energy into the ball, so he did work on it. To do the greatest possible amount of work, he applied the greatest possible force over the greatest possible distance.

If the motion is in the opposite direction compared to the force, then $$W=-Fd$$; the negative work is to be interpreted as energy removed from the object to which the force was applied. For example, if Superman gets in front of an oncoming freight train, and brings it to a stop, he's decreased its energy rather than increasing it. In a normal gasoline-powered car, stepping on the brakes takes away the car's kinetic energy (doing negative work on it), and turns it into heat in the brake shoes. In an electric or hybrid-electric car, the car's kinetic energy is transformed back into electrical energy to be used again.

## Homework Problems

aa / Problem 1.

ab / Problem 2.

1. The beer bottle shown in the figure is resting on a table in the dining car of a train. The tracks are straight and level. What can you tell about the motion of the train? Can you tell whether the train is currently moving forward, moving backward, or standing still? Can you tell what the train's speed is?

2. You're a passenger in the open basket hanging under a hot-air balloon. The balloon is being carried along by the wind at a constant velocity. If you're holding a flag in your hand, will the flag wave? If so, which way? (Based on a question from PSSC Physics.)

3. Driving along in your car, you take your foot off the gas, and your speedometer shows a reduction in speed. Describe an inertial frame in which your car was speeding up during that same period of time.

4. If all the air molecules in the room settled down in a thin film on the floor, would that violate conservation of momentum as well as conservation of energy?

5. A bullet flies through the air, passes through a paperback book, and then continues to fly through the air beyond the book. When is there a force? When is there energy?

6. (a) Continue figure l farther to the left, and do the same for the numerical table in the text.
(b) Sketch a smooth curve (a parabola) through all the points on the figure, including all the ones from the original figure and all the ones you added. Identify the very top of its arc.
(c) Now consider figure k. Is the highest point shown in the figure the top of the ball's up-down path? Explain by comparing with your results from parts a and b.

7. Criticize the following statement about the top panel of figure c on page 42: In the first few pictures, the light ball is moving up and to the right, while the dark ball moves directly to the right.

8. Figure ac on page 60 shows a ball dropping to the surface of the earth. Energy is conserved: over the whole course of the film, the gravitational energy between the ball and the earth decreases by 1 joule, while the ball's kinetic energy increases by 1 joule.
(a) How can you tell directly from the figure that the ball's speed isn't staying the same?
(b) Draw what the film would look like if the camera was following the ball.
(c) Explain how you can tell that in this new frame of reference, energy is not conserved.
(d) Does this violate the strong principle of inertia? Isn't every frame of reference supposed to be equally valid?

ac / Problem 8.

9. Two cars with different masses each have the same kinetic energy. (a) If both cars have the same brakes, capable of supplying the same force, how will the stopping distances compare? Explain. (b) Compare the times required for the cars to stop.

10. In each of the following situations, is the work being done positive, negative, or zero? (a) a bull paws the ground; (b) a fishing boat pulls a net through the water behind it; (c) the water resists the motion of the net through it; (d) you stand behind a pickup truck and lower a bale of hay from the truck's bed to the ground. Explain. [Based on a problem by Serway and Faughn.]

11. Weiping lifts a rock with a weight of 1.0 N through a height of 1.0 m, and then lowers it back down to the starting point. Bubba pushes a table 1.0 m across the floor at constant speed, requiring a force of 1.0 N, and then pushes it back to where it started. (a) Compare the total work done by Weiping and Bubba. (b) Check that your answers to part a make sense, using the definition of work: work is the transfer of energy. In your answer, you'll need to discuss what specific type of energy is involved in each case.

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

##### Footnotes
[1] Actually these statements are both only approximately true. The moon's orbit isn't exactly a circle.
[2] This is really only an estimate of the average force over the time it takes for the bullet to move down the barrel. The force probably starts out stronger than this, and then gets weaker because the gases expand and cool.
[3] This definition is known as Newton's second law of motion. Don't memorize that!
[4] This is called Newton's third law. Don't memorize that name!
[5] During the Scopes monkey trial, William Jennings Bryan claimed that every time he picked his foot up off the ground, he was violating the law of gravity.
[6] This follows from the additivity of forces.
[7] Its initial speed is 0, and its final speed is 10 m/s, so its average speed is 5 m/s over the first second of falling.
[8] “Black box” is a traditional engineering term for a device whose inner workings we don't care about.
[9] For conceptual simplicity, we ignore the transfer of heat energy to the outside world via the exhaust and radiator. In reality, the sum of these energies plus the useful kinetic energy transferred would equal $$W$$.