Question

A body of mass $m$ is accelerated to velocity $v$ in time $t^{\prime }$. The work done by the force as a function of time $t$ will be

• A. $\frac{m}{2}\frac{v^2{t}^2}{t^{\prime 2}}$
• B. $\frac{1}{2}{\left(\frac{mv}{t^{\prime }}\right)}^2{t}^2$
• C. $\frac{m}{2}\frac{v}{t^{\prime }}{t}^2$
• D. $\frac{mv{t}^2}{2t^{\prime }}$

Answer 1

The correct option is A $\frac{m}{2}\frac{v^2{t}^2}{t^{\prime 2}}$

The force as a function of time $t$ on body of mass $m$ is
$F=m.a$ , $a$-acceleration
And the work done by the force as a function of time $t$ is
$W=F.d$ , $d$-distance covered
$W=ma.d$
Here, the body is accelerated to velocity $v$ in time $t^{\prime }$, hence
$a=\frac{v}{t^{\prime }}$
distance traveled $d={\int }_0^t\frac{v}{t^{\prime }}tdt=\frac{v}{t^{\prime }}\frac{t^2}{2}$

$W=ma.d=m\frac{v^2}{{t^{\prime }}^2}\frac{t^2}{2}$