Bicyling Speed

I think I would be an easy target for Nerd Sniping.  Today I was waiting to cross Volunteer Blvd to go yoga when I saw a bicylist start from a stop.

Expected Profile

I started to think about the velocity profile.  From experience I thought it would be something like $1-e^{-x}$ (shown on the left), but I wondered if this could be derived.

In my first model I assumed a constant force, $F=C$.  Acceleration is the derivative of velocity, and force is related to acceleration by Newton's Laws, so with a bit of math I found that the velocity profile for a constant force: $v(t)=C/m t$, where m is the mass.  This linear profile didn't fit in with what I observed and what you all know: it is much easier to peddle a bike after you get it going, but hard to start. (A linear profile allows for this, but not to the extent that is observed.)

Oscillating Profile

I then reconsidered my assumption that the force is constant. A small correction would be to add an oscillating term to represent that it is easier to push down on the peddles when they are upright compared to horizontal, so the force looked like $F(t)=C_1 + C_2 cos[\omega t]$.  This, however, would only make velocity profile oscillate and didn't give me the correct long term behavior.

I added an exponential term to the force $C_1+C_2 cos[\omega t]+C_3 e^{t}$, and that seemed to do the trick, as shown on the left.  I am not quite sure on how to justify the addition of the exponent term, however.  I think an argument could be based on that the amount of force depends on what is already inputed, similar to how atomic decay rates are exponential because they depend on the amount of matter at that time.  Not entirely convinced though, I  get the feeling that I am trying to make the physics fit the model.