The work done in time ‘t’ on a body of mass ‘m’ which is accelerated from rest to a speed ‘v’ in time$\text{'}{t}_1\text{'}$ as a function of time ‘t’ is given by

A body of mass m at rest is accelerated for $t_1$  seconds

Until its final velocity is v

Acceleration $=\frac{v-0}{t_1}$

$a=\frac{v}{t_1}$

Force applied $F=ma$

$\Rightarrow F=\frac{mv}{t_1}$

Displacement produced ‘s’ is expressed as function of time t

$s=ut+\frac{1}{2}a{t}^2$                                    $\because u=0,a=\frac{v}{t_1}$

$s=\frac{1}{2}\frac{v}{t_1}{t}^2$

Hence work done $W=F\times S$

$W=\frac{mv}{t_1}\times \frac{1}{2}\frac{v}{t_1}{t}^2$

So correct option is (d).