Question

A body of mass 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 seconds after the start of the motion is:

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

Answer 1

The correct option is A $\frac{1}{8}m{a}^4{t}^2$

Velocity of body $v=a\sqrt{s}$

acceleration $a^{\prime }=\frac{dv}{dt}=\frac{a}{2\sqrt{s}}\frac{ds}{dt}=\frac{a^2}{2}$

displacement $s^{\prime }=\frac{1}{2}{a}^{\prime }{t}^2=\frac{1}{2}\frac{a^2}{2}{t}^2$

Work done $W=force\times displacement$

$W=m{a}^{\prime }\times s^{\prime }=\frac{1}{8}m{a}^4{t}^2$