A locomotive of mass m starts moving so that its velocity varies according to the law v = alpha sqrt(s), where alpha is a constant and .s. is the distance covered. Find the total work done by all the forces on the locomotive during first .t. second after the begining of motion.

A locomotive of mass m starts moving so that its velocity varies accor

1.	A locomotive of mass m starts moving so that its velocity varies according to the law v = alpha sqrt(s), where alpha is a constant and .s. is the distance covered. Find the total work done by all the forces on the locomotive during first .t. second after the begining of motion.
Answer» SOLUTION :Given, velocity `v = alpha s^(1//2)` Acceleration, `a = (dv)/(dt) = (d)/(dt) (alpha s^(1//2)) = (1)/(2) alpha s^(-1//2) ((ds)/(dt)) = (1)/(2) alpha s^(-1//2) v = (1)/(2) alpha s^(-1//2) (alpha s^(1//2)) = (alpha^(2))/(2)` Now force, `F = ma = (m alpha^(2))/(2)`. Distance covered by locomotive in first t SECOND `s = ut + (1)/(2) at^(2) = 0 + (1)/(2) ((alpha^(2))/(2))t^(2) = (alpha^(2)t^(2))/(4)`, Work done, `W = FS = (m alpha^(2))/(2) xx (alpha^(2)t^(2))/(4) = (m alpha^(4) t^(2))/(8)`.