Question

A body of mass m starts moving from rest along x-axis so that its velocity varies as $v=a\sqrt{s}$ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

• A. $\frac{1}{8}m{a}^4{t}^2$
• B. $8m{a}^4{t}^2$
• C. $4m{a}^4{t}^2$
• D. $\frac{1}{4}m{a}^4{t}^2$

Answer 1

Answer: (A)

Velocity of the body is given by
$v=a\sqrt{s}$
Differentiating w.r.t. $t$, we get
Acceleration, $a^{\prime }=\frac{dv}{dt}=a\frac{1}{2}{s}^{-1/2}\cdot \frac{ds}{dt}=a\frac{1}{2\sqrt{s}}\cdot v$ $\Rightarrow a^{\prime }=\frac{a}{2\sqrt{s}}\cdot a\sqrt{s}=\frac{a^2}{2}$
Force on the body is $F=m{a}^{\prime }=\frac{m{a}^2}{2}$
Distance covered by the body is given by
$s=ut+\frac{1}{2}{a}^{\prime }{t}^2$
$\Rightarrow s=\frac{1}{2}\cdot \frac{a^2}{2}{t}^2$
Work done =Force $\times$ Distance $=\frac{m{a}^2}{2}\cdot \frac{1}{2}\frac{a^2}{2}{t}^2=\frac{1}{8}m{a}^4{t}^2$