How fast can a cyclist accelerate from a standing start?

Question

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.

Answer 1

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 "gain ratio" 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 ("1 rep max") 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.

Answer 2

This image depicts a SWAG at the line between the center of gravity of cyclist + bike and the center of force propelling the bike. (Actually, the center of force would probably be the point of contact for the rear tire, but I'd already drawn this picture when that occurred to me.)

Force will be divided between that used to propel the bike forward and that used to rotate the bike+rider upward. The upward force would be proportional to the sine of the angle relative to ground, which I make out to be around 60 degrees in picture, yielding a sine value of around 0.87. The forward force would be proportional to the cosine of the angle -- about 0.5.

To keep the front tire on the ground it's necessary to keep the upward force below 1g.

So, if the upward force needs to stay below 1, the forward force needs to stay below (1/0.87) times 0.5, or about 0.57.

So the maximum practical acceleration would be about 0.57 times G, or around 5.8 m/s/s.

I suspect that bikes designed for "drag racing" have a longer rear triangle and a more prone rider position to reduce depicted the angle (to maybe 30-45 degrees) in order to make higher acceleration practical.

Answer 3

Assuming that the bike is built for the purpose of achieving maximum possible instantaneous acceleration, there is exactly one limiting factor: The grip of the rear tire on the road.

Force on the pedals is not a limiting factor because you allow for optimal gearing. In theory, you could start with a vanishingly small gear from resting, and thus put an arbitrarily high torque on the rear wheel.

Geometry is not a limiting factor because you can build the bike to have the rear wheel in just the right place, such that the entire weight is on the rear wheel during the acceleration.

But the material of the rear tire and the road surface determine how much horizontal force can be transmitted with a given weight.