Distance as a Function of Speed at Constant Power


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.


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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