Question

A particle of mass $m$ moves on a straight line with its velocity varying with the distance travelled according to the equation $v=\alpha \sqrt{x},$ where a is a constant. Find the total work done by all the forces during a displacement from $x=0$ to $x=d$.

• A. $\frac{1}{2}m{a}^{2}b$
• B. $\frac{1}{4}m{a}^{2}b$
• C. $m{a}^{2}b$
• D. $2m{a}^{2}b$

Answer 1

Answer: (A)

$\frac{1}{2}m{a}^{2}b$

$v=\alpha \sqrt{x}$

$⟹\frac{dx}{dt}=\alpha \sqrt{x}$

$⟹2\sqrt{x}=\alpha t$

$⟹x=\frac{{\alpha }^2{t}^2}{4}$

Work done by forces$=$${\int }_0^{b}Fdx{=}_0^b\int madx$

${=}_0^b\int m\frac{d^{2}x}{d{t}^2}dx$

${=}_0^b\int m\left(\frac{2{\alpha }^2}{4}\right)dx$

$=\frac{1}{2}m{\alpha }^{2}b$