The problem was based on a real-life scenario.
First let’s find the position of something that accelerates with constant power, starting at rest.
Its kinetic energy is
with the power, the kinetic energy, and the time it’s been accelerating. Kinetic energy is , so
or solving for
Constant power is somewhat less effective than constant acceleration.
The time needed to complete a race, accelerating the whole way under constant power, is
So the time scales with the minus one-third power of the power output. However, if we add in stopping and turning around, this will get even weaker, because it’s significantly easier to turn around when you aren’t already going fast (although the details depend on the length of the race and the coefficient of friction, and I don’t think it’s very enlightening to work out in full detail.)