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*v^{2})/dt =
1/2*m*d(v^{2})/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*v^{2}*dv/ds. That makes P*ds=m*v^{2}*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*v^{3}. So
s=m*v^{3}/(3*P). You can confirm this by taking
the derivative with respect to s of both sides of this
expression.
This information is brought to you by M. Casco Associates, a
company dedicated to helping humankind reach the stars through
understanding how the universe works. My name is James D. Jones.
If I can be of more help, please let me know.

JDJ