Work done in time t on a body of mass m which is accelerated from rest to a speed v in time t as a function of time t is given by

The correct answer is (vt/t₁)²m/2.

To find the work done on the body as a function of time, we can apply the work-energy theorem, which states that the work done on a body is equal to the change in its kinetic energy.

Step 1: Find the acceleration of the body.
The body starts from rest, which means its initial velocity is:
$u=0$
It is accelerated to a speed $v$ in time $t_1$. Assuming the acceleration $a$ is uniform, we use the first equation of motion:
$v=u+a{t}_1$
Substituting $u=0$:
$v=a{t}_1$
Solving for acceleration $a$:
$a=\frac{v}{t_1}$

Step 2: Find the velocity at any time t.
The velocity $v_t$ of the body at any time $t$ is given by:
$v_t=u+at$
Since $u=0$, substituting the value of $a$ gives:
$v_t=\left(\frac{v}{t_1}\right)t=\frac{vt}{t_1}$

Step 3: Calculate the work done in time t.
According to the work-energy theorem, the work done $W$ in time $t$ is equal to the kinetic energy acquired by the body in that time:
$W=\frac{1}{2}m{v}_t^2$
Substituting the expression for $v_t$:
$W=\frac{1}{2}m{\left(\frac{vt}{t_1}\right)}^2$
Rearranging the terms gives the final expression:
$W={\left(\frac{vt}{t_1}\right)}^2\frac{m}{2}$