# Biking at (half) the speed of Light

Today a thought occurred to me as I was biking to campus; you could never bike at the speed of light.

Okay, this seems pretty obvious, and if you didn’t know I was an aerospace engineer, you’d probably think I was an idiot (maybe you still do). It would take a lot of leg power to even achieve an admirable fraction of the speed of light on a bike, even without friction! However, if you think about the problem statement as a physics application rather than a super-Lance Armstrong-esque athletic test, then you can see where my dilemma started.

A bicycle operates with a frame mounted upon two parallel wheels, which rotate upon axles to give you a velocity in the direction desired. But you already knew that. What fewer people realize is that when your bike is in motion, different parts of the wheel have different velocities with respect to the earth.

If you spin a suspended wheel, the velocity of every component of the wheel is the radius multiplied by the radial velocity in the direction of rotation. However, when that wheel is placed upon the ground with friction, those components “stick” to the surface of the Earth upon contact, then rise up and fly over the Earth as the wheel spins.

If you punch in the numbers, the velocity of the bottom of a non-slipping wheel in motion is always 0*v, where ‘v’ is the velocity of the bike frame, and the velocity of the top of the wheel is 2*v.

This is when my mind started to wander. If the top of the wheel’s velocity is 2*v, then according to the laws of physics, you could never ride a bike faster than 0.5*c! The frame of the bike would be traveling 0.5*c while the bottom of the wheel would be travelling 0 and the top of the wheel would be travelling c.

I felt so smart.

That is, until I remembered relativistic velocities are not additive. According to Einstein's theory of special relativity;

v’ = (u + v)/(1+uv/c^2)

‘v’ is the velocity that you are currently traveling at. Take that as our bike. It is traveling at 0.5c. ‘u’ is the added velocity. If the top of the wheel is traveling at 2*v, the bike’s velocity, that is an additional ‘v’ that we are adding to the 0.5c. Thus ‘u’ = 0.5c as well. We plug it in and get:

v’ = (0.5c + 0.5c)/(1 + 0.25c^2/c^2)

which simplifies to:

v’ = 0.8c

Alas, the top of the wheel is not traveling the speed of light. And it never will be. Even if we push Lance to the limit of his abilities with all the dope the US can manufacture, the top of the wheel will always be less than the speed of light.

This, however, would probably look pretty cool to a bystander. Because the length of objects contracts as one approaches the speed of light, the wheels would be ‘pear’ shaped. The top of the wheel would appear to contract a lot, being closer to c, and the bottom would be almost perfectly round, traveling 0.

While the bottom of the tire would appear normal, light leaving the top of the tire would be compressed by the Doppler effect. This would make the whole top half of the tire appear a bluish tint, and finally purple by the apex.

Furthermore, time would actually pass slower for an ant stuck on the wheel as it was closer to the top, then speed back up again as the wheel’s velocity reattained 0. It would be a confusing ride.

Finally, the mass of objects increases as they near the speed of light. This means that the mass of the top of the wheel would actually be more than that of the bottom of the wheel, changing the entire center of mass of the bike completely. This throws another wrench in the idea of biking close to relativistic velocities.

Moral of the story: Nothing, not even a bike wheel, can travel faster than the speed of light. Thanks Einstein!