Provided a racing car does not lose traction, the time taken by it to race from rest through a distance depends mainly on engine's power P. Derive the time t in terms of S and P, assuming that power is constant.

Explanation

`P=mav=mv(dv)/(dt)(a=(dv)/(dt))`
`impliesunderset(0)overset(v)intvdv=(rho)/(m)underset(0)overset(t)intdt`
Find v, then
`v=dx/dt`
`underset(0)overset(x)intdx=overset(t)underset(0)intvdt`
Find x.
The power of an engine is the rate at which energy is supplied as given by
`P=mav=mv(dv)/(dt)`
`impliesunderset(0)overset(v)intvdv=P/m underset(0)overset(t)intdt`
`implies(v^(2))/(2)=P/mt`
`impliesv=sqrt((2P)/(m))t^(1//2)`
`impliesunderset(0)overset(t)intdx=sqrt((2P)/(m))underset(0)overset(t)intt^(1//2)dt`
`impliesx=sqrt((8P)/(9m))t^(3//2)`