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

Options

A. `8 ma^(4)t^(2)`
B. `1/4ma^(4)t^(2)`
C. `4 ma^(4)t^(2)`
D. `1/8ma^(4)t^(2)`

Answer

C

Explanation

`v = asqrt(s) = (ds)/(dt)`
`S^(-1//2) = ds = adt`
`rArr " " int_(0)^(S)S^(-1//2) ds = a int_(0)^(t)dt rArr sqrt(s) = (at)/2 " "…(i)`
`v=asqrt(s) rArr v = (a^(2)t)/2 " "…(ii)`
At `t = 0 rArr v = 0 rArr K_(1)=0`
At time `t rArr v = (a^(2)t)/(2 ) rArr K_(2) = 1/2 mv^(2) = 1/8 ma^(4)t^(2)`
Work done by all the forces is equal to change in K.E
`W =K_(2)-K_(1) = 1/8 ma^(4)t^(2)`