An automobile of mass m accelerates from rest. If the engine supplies a constant power P, the velocity at time t is given by -

Explanation

Velocity: As `Fv=P=cont ant`, i.e.,
`m(dv)/(dt)v=P[as F=(mdv)/(dt)]`
or `int vdv=int (P)/(m)dt`
By integrating both sides , we get `v^2/2=P/mt+C_1`
As initially the body is at rest, i.e. `v=0` at `t=0`, so `C_1=0`
`:.v=((2Pt)/(m))^(1//2)`
Position: From the above expression, `v=((2Pt)/(m))^(1//2)`
or `(ds)/(dt)=((2Pt)/(m))^(1//2)` [ As `v=(ds)/(dt)`]
i.e., `intds=int((2Pt)/(m))^(1//2)dt`
By integrating both sides, we get
`s=((2P)/(m))^(1//2)*(2/3)t^(3//2)+C_2`
Now as at `t=0`, `s=0`, so `C_2=0`.
`s=((8P)/(9m))^(1//2)t^(3//2)`