If a car is driven along a straight path by an engine of constant power , find the displacement -time relation.

Explanation

Let the power of the engine be P (constant).
But P= velocity `xx` force = `vma=vm (dv)/(dt) or , vdv = P/m dt.`
Integrating `(v^2)/(2) =P/m t+C` where C = integration constant.
`or, v=sqrt(2)sqrt(P/mt+C)`
`or, ds=sqrt(2) sqrt(P/mt+Cdt)" "[because v=(ds)/(dt)]`
Let `P/m t+C=z`
`therefore P/m dt=dz or, dt=m/P dz`
`therefore ds =sqrt(2) m/P z^(1/2) dz`
By further integration we get,
`s=(sqrt(2)m)/P 2/3z^(3//2)+C^.`
`=(sqrt(2)m)/P*2/3(P/mt+C)^(3//2) +C^.`
[`C^.` = integration constant ]
`=(2sqrt(2))/(3) sqrt(P/m)(t+(mC)/P)^(3//2) +C^.`
This is the required relationship between time and displacement.