Question:
A particle of mass $m$ moves on a straight line with its velocity varying with the distance travelled according to the equation $v=a \sqrt{x}$, where a is a constant. Find the total work done by all the forces during a displacement from $x=0$ to $x=d$.
Solution:
$\mathrm{v}=\mathrm{a}^{\sqrt{x}}$
$v_{1}=0$ and $v_{2}=a \sqrt{d}$
$a^{\prime}=\frac{\left(v 2^{2}-v 1^{2}\right)}{2 d}=\frac{a^{2}}{2}$
Force $\mathrm{F}=\mathrm{mal}=\left(\mathrm{ma}^{2}\right) / 2$
Work done $\mathrm{W}=\mathrm{Fd} \cos \Theta$
$\mathrm{W}=\frac{\frac{m a^{2} d}{2}}{2}$