Power required to accelerate an object question?

[olso4051]

I want to calculate the mechanical power I can output continuously on a stationary bike. The good news is I have the power vs. speed curve available for the given bike, the question is how much power does it take to accelerate from one speed to another?

If I know it takes 250W at 20mph and 275W at 21mph, how many watts does it take to accelerate from 20mph to 21mph in 1 second? I think I need to know something about the inertia of the system but that could be calibrated if I just knew the equations.

Also how would the equations change if it was not a stationary bike assuming riding in a vacuum on flat ground?

 

[mfb]

The bike has some mass which has to be accelerated - for a regular bike, this is mainly the rider and the bike itself, for a stationary bike I think it is some rotating disk.
Anyway, this has inertia, and a corresponding energy for each velocity. You can take the difference between (energy hat 21mph) and (energy at 20mph) and get the energy required for the acceleration. Power is now simply this energy divided by the acceleration time.

It is possible to estimate those numbers without analyzing the bike itself: Ride with 21mph, reduce your power to 0 and measure the time until the bike is at 20mph. The bike will continue to consume 250-275W, and take this power from its inertial energy.

All bikes can be described with the same procedure (unless you introduce some very strange active constructions). The inertial energy is proportional to the velocity squared. The power required to hold a constant speed depends on the velocity - this will be different for stationary and moving bikes, but as you already know it this is not important here.