(Or, how to tell how fast you’re going based on how much mud is hitting you in the face.)

I was riding my bike in the rain yesterday, along a dirt road, listening to Let it Go from Frozen, when I realized that if I leaned forward I started getting a lot of mud in my face from water kicked up off my tire. That water gets flung off the tire at whatever the rotational speed of the wheel is. The faster I go, the higher the water goes. Maybe I could use that to tell how fast I was going. (I have a bike speedometer, but I still haven’t installed it. That was Christmas two years ago so it might never happen.)

Then I was like, but, I’d have to estimate the size of a water droplet, and if there was dirt in it that would increase the mass, so I’d also have to make some assumptions about percent dirt content.

But, no! The height of the droplet can be calculated with conservation of energy, which means the mass drops out. As long as I’m ignoring air resistance, which is probably ok, I can pretty simply calculate my speed based on how far I have to lean forward before I start getting mud in my face.

Here’s how it works:

As I ride forward, water droplets cling to the surface of the tire. At some point, they are flung off. When they are flung off, they travel in a straight line out from the surface of the tire. They travel tangentially, and their speed is equal to the linear speed of the rotating wheel.

The linear speed of the wheel is equal to the speed that I’m bicycling at. That means that the water droplet should theoretically be travelling at the same speed at which I am bicycling.

Now all that I have to do is use conservation of energy to find that speed, given the height.

There are two types of energy involved, linear kinetic and gravitational potential. Here are the equations:

Here, m is the mass of the object, v is its velocity, h is its height above the surface of the earth in meters, and g is the acceleration due to gravity (approximately 9.8.)

The idea here is that energy is never created or destroyed. It just changes forms. In this case, it changes from kinetic energy to potential energy.

(I’m making several simplifications here. I’m assuming no air resistance, and no energy lost to deformation of the water droplet.)

Ok, here’s the calculation:

This equation at the end you can use to find your velocity. Just plug in the height your head is above the wheel when you start getting mud in your face.

Now for some numbers.

My usual head height is 3 feet, 3 inches above the midline of my wheel (I’m making another simplification, assuming that my head is directly vertical above the back edge of the wheel:

I’ll need to convert that to meters. For that I’ll just use google’s handy conversion feature.

Apparently 3’3″ is .9906 m

Plug this in to my equation:

And convert to miles per hour:

Here’s a table of head heights versus bike speed:

The cool thing is that it doesn’t matter how big your tire is, or how big the water droplets are, or how much dirt the droplets have in them. All that matters is how fast you’re going and how high above the wheel your face is.

There is probably a cutoff point, where if you’re going slower than that the water isn’t lifted at all, and of course if you’re going too fast you won’t be tall enough to reach the height of the water. Also, you can’t have fenders on your bike.

But, that’s how you can tell how fast you’re going based on how much mud you get in the face!