# Momentum

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$ g / It doesn't make sense to add his debts to her assets. h / 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. i / A ball is falling (or rising). j / The same ball is viewed in a frame of reference that is moving horizontally. k / The drops of water travel in parabolic arcs. m / 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. n / 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 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.

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.

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 -0.5 [/itex] +0.5 [/itex] big ball +1 [/itex] +0.5 [/itex]

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

momentum = mvis conserved.

This is our second example of Noether's theorem.

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

• Is momentum conserved in figure c/1?
• We have to check whether the total initial momentum is the same as the total final momentum.

Yes, momentum was conserved:

0+mv = mv+0

#### Example 3: Ice skaters push off from each other Figure $$\PageIndex{1}$$: Figure d: Example 3.

Let's check that this is consistent with conservation of momentum:

Momentum was conserved:

0+0 = m×(-v)+mv

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:

#### Example 4: Unequal masses

• 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?
• 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.

## 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 Figure $$\PageIndex{1}$$: Figure / Example 5.

#### 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. f / Example 8.

#### Example 9: Momentum and kinetic energy of a glacier

A cubic-kilometer glacier would have a mass of about 1012 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 kg⋅m/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 kgm/s [/itex] force newtons (N)

#### Example 10: A bullet

• A bullet emerges from a gun with a momentum of 1.0 kg⋅m/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?
• The force is2

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

• 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.
• 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

Equations
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

equations

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.

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, 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}{

 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 28, 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 l, 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. l / Example 12.

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 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.