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# 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.

### 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

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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.

### Homework Problems

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?

y / Problem 1.

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.)

z / Problem 2.

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 j 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 i. 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 44: 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 aa on page 62 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.

1. How can you tell directly from the figure that the ball's speed isn't staying the same?
2. Draw what the film would look like if the camera was following the ball.
3. Explain how you can tell that in this new frame of reference, energy is not conserved.
4. Does this violate the strong principle of inertia? Isn't every frame of reference supposed to be equally valid?

aa / 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.

### References

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.