Question

The velocity $\left(v\right)$ 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 $x=2m$.

• A. $m{\alpha }^2$
• B. $m\alpha$
• C. $\frac{m\alpha }{2}$
• D. None of these

Answer 1

The correct option is A $m{\alpha }^2$

$v=\alpha \sqrt{x}$
$\frac{dx}{dt}=\alpha \sqrt{x}$
$\Rightarrow {\int }_0^x\frac{dx}{\sqrt{x}}={\int }_0^t\alpha dt$
$x=\frac{{\alpha }^2{t}^2}{4}$
$v=\frac{{\alpha }^{2}t}{2}$
$a=\frac{{\alpha }^2}{2}$
$F=ma=\frac{m{\alpha }^2}{2}$
$W=F\cdot \Delta x=\frac{m{\alpha }^2}{2}\times 2=m{\alpha }^2$