## Reflections on the Moon

Q: Why don’t elephants play tennis?
A: They prefer squash.

Two players compete in a fast-paced game at the gym. I exercise across from them, watching as they smash a blue rubber ball in turns. The game is in a small indoor court, and the ball moves very quickly. Its motion, followed from a distance, appears nearly rectilinear. All of the walls, the floor, and the ceiling are in play, and the ball’s wild meanderings, jots of wild color, mesmerize me as I stretch.

The game is squash (Wikipedia), which I had heard of but not seen played before. Watching the squash players, I was at first surprised by their ability to predict the seemingly-chaotic motion of the ball. However, a geometric property of the court aids them.

The squash ball moves very quickly (faster than a major-leaguer’s fastball), so over the short distances of the court, we can approximate its motion as roughly a straight line. The court is a rectangular prism, and this shape has the property that if a player smashes the ball at a corner, the ball will pop back out right at them, parallel to the way it came in. In two dimensions this is shown in the following diagram:

The squash ball follows the dark blue path, bouncing off the corners at points A and B. The light blue line shows the continuation of the paths, with a hypothetical meeting point D if the incoming and outgoing lines are not parallel.

The ball comes into the corner bouncing at point $A$, making an angle $\theta$ with the wall. By hypothesis, it bounces off with the same angle, then comes into the next wall with an angle $\phi$ at point $B$. It bounces off with this angle as well and returns to the court. We’re trying to show that the incoming and outgoing paths are parallel.

To do this, I’ve drawn in light blue the continuation of the incoming and outgoing paths. If they’re parallel, they never meet, and the angle drawn as $\delta$ should be zero. Notice that $\theta$ and $\phi$ are the small angles of a right triangle $ABC$, so they add to a right angle. Angle $CBD$ is opposite $\phi$, and so equal to it. That means angle $ABD$ is $2\phi$ and similarly angle $BAD$ is $2\theta$. Those two angles, along with $\delta$ form the triangle $ABD$. However, they add to a straight angle by themselves, and so we must have $\delta = 0$, showing that the ball pops out parallel to the way it came in, allowing the players to predict its motion easily.

In three dimensions, this is just the same, except that you have to work through the argument over again. A student of mine pointed out a different argument to come to the same result. Set up and $x-y-z$ coordinate system at the corner along the intersections of the planes. Then one wall works by flipping the $x$-coordinate of the incoming ball’s velocity vector, and the other two flip the ball’s $y$ and $z$-coordinates, so that after bouncing off all three, the velocity vector is reversed.

In squash, this result is far from perfect because gravity affects the ball’s motion, and its rotation, along with friction from the walls, may affect its angle of reflection. Energy is lost in each reflection as well, and the ball will slide some against the wall, so all in all it’s a rough approximation.

For light the approximation is much better as long as the wavelength of light is much smaller than the size of the mirrors. The setup with three orthogonal plane mirrors like this is a called a retroreflector because it reflects light back the way it came. If you look into one from any angle, you will see your own pupil at the corner, because at the corner the incoming and outgoing rays are not only parallel, but on top of one another, so light must start and end at the same place after reflecting there. All the light you see ends at your pupil, so that’s what you see in the corner. If you have three hand mirrors, it’s an easy experiment to try.

One interesting application of this idea is shown here:

A retroreflector on the moon's surface. The Apollo missions left this array of hundreds of reflectors intended for extremely accurate measurements of the Eart-Moon distance.

This is a retroreflector array on the moon. When an Earth-based laser sends a pulse of light at the moon, the retroreflectors send the pulse back to Earth. If you could measure the trip time very precisely, you can multiply by the speed of light to find the distance to the moon. The APOLLO project (not the lunar orbiters, but the ground-based Apache Point Observatory Lunar Laser-ranging Operation) is trying to do this to an accuracy of one millimeter.

I’ve sometimes heard silly things like “the Campbell’s soup cans thrown out by Americans in a single month could stretch to the Moon and back three times.” I say this is silly because

• Why would you want to do that?
• I totally just made the statistic up because it is meaningless and nonmemorable. Things like per-capita consumption, percentages of usable land being turned into dumps, and statistics about ecological impact actually mean something.
• No they can’t. They would fall down if they tried.

Regardless, if you know the distance to the moon with one millimeter accuracy, then your estimate of how many Campbell’s soup cans away it is is limited by how accurately you know the size of a Campbell’s soup can until you measure the can to an accuracy of single atom (and Campbell’s soup cans vary from one to another by a lot more than that, and don’t stack perfectly regularly, and shift around, and get hotter and colder, etc).

There are a few good reasons you’d want to know the position to the moon so precisely. Perhaps the most striking is as an extremely tight test of the equivalence principle of general relativity. The Earth and Moon have different densities, and so might conceivably fall towards the sun at different accelerations, even when they’re the same distance away. Modern cosmology and theoretical physics frequently explore theories of modified gravity in attempts to explain the acceleration of the universe’s expansion or create quantum theories of gravity. If the equivalence principle doesn’t hold, watching the acceleration of the moon very closely could be the first place we’d get a hint of it.

### 3 Responses to “Reflections on the Moon”

1. Paul James Says:

Very interesting (haha and entertaining) information about the earth to moon based laser measurement system.

2. Matthew Says:

Very nice, except that squash balls don’t bounce off side walls at the same angle they hit them. When you start playing squash the bounce off the side wall is mystifying because instinctively you expect it to bounce off at the same angle. In fact the ball seems to pop out closer to perpendicular to the wall, usually making it easier to hit.

3. Mark Eichenlaub Says:

Yes, you’re right. I believe I discussed in the post that the model is not correct for actual squash balls.