[jyl]

You guys are helping me with the intuition.

Here's where my hangup has been. Suppose I have a spaceship of mass M with a rocket drive that produces a constant force F. It is far from any planet or sun, so there is no friction opposing its movement. Force F accelerates the spaceship to velocity V at time T1, then a while later it has accelerated to velocity 10*V at time T2. At time T1 the power of the drive is F x V. At time T2 the power is F*10*V. I was/am having trouble understanding intuitively why the constant force is producing more and more power.

Mathematically I think I get it. At time T the ship has kinetic energy KE = 0.5*M*V^2. After a small increment of time dT, at time T+dT it has KE = 0.5*M*(V+F/M*dT)^2. The change in kinetic energy is 0.5*M*[2*V*F/M*dT + (F/M*dT)^2]. The average power during dT is change in kinetic energy divided by dT, or 0.5*M*2*V*F/M*dT/dT + 0.5*M*(F/M)^2*dT). dT/dT = 1 in the first term, and as dT goes to zero, the second term goes to zero, so the instantaneous power P = 0.5*M*2*V*F/M = F*V.

Right?. That's question 1.

(This illustrates, at an elementary level, why I abandoned my physics major and switched to a math major, back in school. I could do the math but lacked intuition for the physics. What good is a physicist who can't intuitively grasp power?. But that was many decades ago and now I don't grasp any of it.)

So here is my further question 2. I assumed that, to hold the 2 kg weight at 3 oclock, the mechanism in the rim has to deliver a constant force of 20 N to the weight, thus a constant reaction force of 20 N is being applied to the wheel. That's why I thought of this device as equivalent to a pulley with a 2 kg hanging weight. Is that correct?. Or does the weight need increasing force to hold it at 3 oclock, as the wheel rotational speed increases?

Finally, question 3. It seems this would be a cheating bicycle motor (yup, that's the intended hypothetical purpose) that would be of very little help at low speeds (when it is delivering very little power), would have very little ability to help the rider make a sharp acceleration at any speed (because its maximum force is quite small), but would be very helpful if the rider is trying to maintain a high constant speed (when it is delivering high power). Is that correct?