Q: A machine delivers power to a body which is directly proportional to velocity of the body. If the body starts with a velocity which is almost negligible, find the distance covered by the body in attaining a velocity v .

Sol: $\large P \propto v $

$\large P = C v $ ; where C = constant

$\large m v \frac{dv}{dx} v = C v $

$\large m v \frac{dv}{dx} = C $

$\large \int_{0}^{v} v dv = \frac{C}{m} \int_{0}^{x} dx $

$\large \frac{v^2}{2} = \frac{C}{m} x $

$\large x = \frac{1}{2} \frac{m v^2}{C}$