The velocity $(v)$ of a particle of mass $m$ moving along x-axis is given by $v = \alpha \sqrt {x}$, where $\alpha$ is a constant. Find work done by force acting on particle during its motion from $x = 0$ to $x = 2m$.

$v=\alpha\sqrt{x}$

$\cfrac{dx}{dt}=\alpha\sqrt{x}$

$\Rightarrow \int_{0}^{x}{\cfrac{dx}{\sqrt{x}}}=\int_{0}^{t}{\alpha dt}$

$x=\cfrac{\alpha^2t^2}{4}$

$v=\cfrac{\alpha^2t}{2}$

$a=\cfrac{\alpha^2}{2}$

$F=ma=\cfrac{m\alpha^2}{2}$

$W=F\cdot \Delta x=\cfrac{m\alpha^2}{2}\times 2=m\alpha^2$