Distance as a Function of Speed at Constant Power

Question:

A car has mass m and accelerates along a straight horizontal test track from rest such that the power is always a constant amount, P. Determine how far the car must travel to reach speed, v.

Answer:

Power is the rate of change of work with respect to time and the change in work is the change in kinetic energy so power = d(KE)/dt = d(1/2*m*v²)/dt = 1/2*m*d(v²)/dt = 1/2*m*2*v*dv/dt. But dv/dt=ds/dt*dv/ds and ds/dt=v, so dv/dt=v*dv/ds making P=m*v²*dv/ds. That makes P*ds=m*v²*dv. Since power is constant, the integral of the left side is just P*s. The integral of the right side is 1/3*m*v³. So s=m*v³/(3*P). You can confirm this by taking the derivative with respect to s of both sides of this expression.

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