Contents
Section 2.1 - Translation Symmetry
Section 2.2 - The Principle of Inertia
Section 2.3 - Momentum
Section 2.4 - Newton's Triumph
Section 2.5 - Work

Chapter 2. Conservation of Momentum

Physicist Murray Gell-Mann invented a wonderful phrase that has since entered into popular culture: “Everything not forbidden is compulsory.” Although people now use it as a sarcastic political statement, Gell-Mann was just employing politics 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

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.

2.3 Momentum

Conservation of momentum

Let's return to the impossible story of Jen Yu and Iron Arm Lu on page 39. 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.

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.

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:

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.

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

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

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.

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

Force

Definition of force

When momentum is being transferred, we refer to the rate of transfer as the force.² 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:

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.

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

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 d on page 46 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 g. 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 h, 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:

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

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,

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

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

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 i 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.

Now let's look at the ball's motion in a new frame of reference, j, which is moving at 10 meters per second to the left compared to the frame of reference used in figure i. 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}{

}

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.

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 m 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 o 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 n 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.

o / 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 p 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 p/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, p/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.

p / 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

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 q, 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.

q / 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:

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 T² is proportional to r³, 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.

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 5 of my online book Conservation Laws.

v / 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 v, 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

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.

x / 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 8.