A fly has been buzzing around my office room for the last couple of days. It navigates its way from corner to corner, lamp to telephone, computer monitor to my nose, and from window to mirror. It astutely steers its way around chairs and speakers and bedposts and guitars. It often descends quickly and appears to instantaneously land on some object. Watching it for a while, the fly seems pretty smart. But then, without warning, it will, with great speed, crash head-first into the window or the mirror.
Two things are odd about this behavior.
1. Why, when advancing directly into the mirror, the fly doesn’t see its reflection, think there is another fly coming right at it, turn to avoid it, and avert the crash into the mirror.
2. Why, after smashing its head into the mirror once, the fly continues to smash its head into the mirror over and over and over again.
Although I am curious about the answers to both these questions, I am more interested in why the head-crash into the mirror does not flatten the fly’s face.
My plan is to devise an experiment from which I will use the quantitative results to mathematically derive the solution to this “non-flattened fly’s face” mystery (nf^3 for short).
I believe the mathematics will involve the concepts of
1. momentum, which involves the fly’s mass and velocity
2. impulse, which involves the change in momentum
3. conservation of momentum, which involves external and internal forces acting on the fly
4. coefficient of restitution, which is the tendency of a body to return to its normal shape once it has been deformed, which involves the elasticity of the fly’s head.
I hope to obtain the pertinent values through an internet search.
I tried to obtain the linear velocity of a fly experimentally, but without success.
The following video shows the result of my attempt to measure a fly’s linear speed.
I snatched at fly off a loaf of bread I keep by my computer to snack on. See Figure 1.

Figure 1 Loaf of Bread
I took it outside to the street where I had set up a 1000 foot linear trajectory which I had fitted with two atomic clock to measure the fly’s time over the 1000 foot linear course. See Figure 2.

Figure 2
The experiment did not go as planned. Although I thought I made it clear to the fly that it was to proceed directly in a straight path from Clock A at the beginning of the course to Clock B at the end of the course, it did not do that. It veered off to the left into the bushes and disaster. See Figure 3 and the video which follows it.

Figure 3
That experiment didn’t work, so I checked for fly flying speeds on the internet. I found various speeds in a classical calculus problem. See Problem 1.
Problem 1: Two trains 300 miles apart are traveling toward each other along the same track. One train travels at 60 miles per hour and the other train at 90 miles per hour. A fly buzzes back and forth from the first train to the second train until the trains collide. If the fly’s speed is 20 miles per hour, how far will the fly travel?
This problems lists the speed of the fly at 20 mph, while other versions of the problems listed speeds of 60 mph, 100 mph, and even 120 mph.
I choose to use the more reasonable 20 mph speed. For my computations, I will assume the fly crashes head-first into the mirror at 20 mph. My question is why, flying directly into the inelastic surface of a glass mirror, the fly doesn’t bounce off with a flat face?
I posed this question to two of my mathematician friends, Guy and Wade. They both agreed that crashing into a hard surface at 20 mph would flatten anything’s face. See the Before and After pictures of Guy and Wade in Figures 4 and 5.

Figure 4 Guy and Wade before demonstration

Figure 5 Guy and Wade after demonstration
They are both okay now and look just as good as they did before their test crash.
After this clever demonstration, I think it is very possible that fly’s do flatten their heads when they smash into a mirror and maybe that is why they just keep doing it again and again.
Exercise: How far does the fly in Problem 12 travel?
Answer: The fly spends the same amount of time traveling as do the trains. The trains are traveling at 60 mph and 90 mph hour toward each other, so their combined speed is 150 mph. Then, since
distance = rate x time, or

So the trains travel for 2 hours. The fly travels 20 miles in one hour and it travels for 2 hours, so in the 2 hours the fly will travel
miles.