A locomotive of mass m starts moving so that its velocity varies according to the law `V=alphasqrts,` where `alpha` is a constant and s is the distance covered. Find the total work done by all the forces acting on the locomotive during the first t seconds after the beginning of motion.

Explanation

Differentiating `v(s)` with respect to time
`(dv)/(dt)=(a)/(2sqrts)(ds)/(dt)=(a)/(2sqrts)asqrts=a^2/2=w`
(As locomative is in unidirectional motion)
Hence force acting on the locomotive `F=mw=(ma^2)/(2)`
Let, at `v=0` at `t=0` then the distance covered during the first t seconds
`s=1/2wt^2=1/2a^2/2t^2=a^2/4t^2`
Hence the sought work, `A=Fs=(ma^2)/(2)((a^2t^2))/(4)=(ma^4t^2)/(8)`