A particle of mass `m` moves on a straight line with its velocity varying with the distance traveled. Find the total work done by all the forces during a displacement `x=0` to `x=d` if the velocity is equal to (a) `v=lamdasqrtx` and (b) `v=lamdax` is constant.

Explanation

We know `W_(t otal)=K_2-K_1`
(a) `x=0`, `v_1=0`, `K_1=0`
`x=d`,`v_2=lamdasqrtd`,`K_2=(1)/(2)mv_2^2=(1)/(2)=(1)/(2)mlamda^2d`
`W_(t otal)=K_2-K_1=(1)/(2)mlamda^2d`
(b) `x=0`,`v_1=0`,`K_1=0`
`x=d`,`v_2=lamdad`,`K_2=(1)/(2)mv_2^2=(1)/(2)=(1)/(2)mlamda^2d^2`
`W_(t otal)=K_2-K_1=(1)/(2)mlamda^2d^2`