How fast can a cyclist accelerate from a standing start? I'm less interested in the empirical answer to this question and more interested in the theoretical physical dynamics that govern this.
I understand that the rider applies some force to the pedals (which could also be thought of as a torque given the length of the crank arm),but the answer isn't simply F=ma because there are gearing ratios involved. In short, I'd like to know the process of calculations by which one could approach this problem and determine the maximum acceleration of a rider.
I suspect some assumptions will be necessary:

Max force applied to pedals = rider weight (this is probably not quite correct if using clipless pedals but probably fairly close?)
Optimal gearing ratio is available
Let's assume a relatively instantaneous problem regime, i.e. a 2-second standing start or something like that

If more information or assumptions are needed, please advise. I'm also open to migrating this to say SE physics if that's a more appropriate place for it.
The power-drag equation is well-understood if not widely-known, and is discussed in this bicycles.SE answer. In particular, the drag equation shows the component for changes in kinetic energy, viz., acceleration. You're asking about a standing start, so initial speed is low and we can ignore the aerodynamic drag component. In addition, let's assume the rider and bicycle is on a flat surface, so we can also ignore the potential energy component. Finally, although there is a rolling and mechanical resistance component to drag, it is typically small so let's ignore it here.
What's left is the kinetic energy component. F = m a is the correct equation to use, but you have to convert F at the pedal to F available at the rear wheel. Then a = F / m. There are a series of lever arms between the pedal and the point at which the rear wheel touches the ground involving the length of the crank arm, the gear ratio, and the radius of the rear wheel. Let's say the total lever arm &quot;gain ratio&quot; is c. Then F Newtons of force at the pedal translate to F / c Newtons at the rear wheel. If the total mass of bike and rider is m, then the acceleration of the bike and rider is F / (m * c). Martin et al. (2006) uses this technique to model sprint performance, with very good results.
Maximum pedal force can actually be greater than body weight, if you can hold onto the handlebars. In fact, maximum pedal force approaches 1RM (&quot;1 rep max&quot;) force from a leg press. A world-class track sprinter has a 1RM of roughly 4x body mass; more commonly, most of us can produce pedal forces equivalent to at least twice body mass. So, depending on the gain ratio calculated above, maximum acceleration is likely to be somewhere in the ballpark of 0.2g to 0.4g, or between roughly 2 and 4 m/s^2.