Question

A body of mass m is accelerated to velocity $v$ in time $t'$. The work done by the force as a function of time $t$ will be?

Answer 1

Assume that the force which is acting on the body is constant(in magnitude and direction) and the body starts from rest and hence the acceleration which the body experiences is constant and then approaches the problem.

Complete step by step answer:
Step 1: Mass of the body $= m$
After acceleration velocity $= v$
We know that, Work done by the force can be defined as force x displacement.
$W = F.d$ (1)
And force can be defined as mass x acceleration.
$F = m.a$ (2)
So, from equations (1) and (2)
$W = m.a.d$ (3)
Step 2: Given that the body is accelerated to velocity $v$ in time $t'$
So, acceleration = change in velocity/ time taken for the change in velocity
$a = \dfrac{v}{{t'}}$ (4)
Step 3: Distance travelled by the body can be calculated using 2nd equation of motion
$d = ut + \dfrac{1}{2}a{t^2}$; where $u =$ initial velocity i.e. 0 m/s, $d =$distance travelled, $t =$time, $a =$acceleration of the body
$d = \dfrac{1}{2}a{t^2}$ (5)
Step 4: Now work done can be calculated from equations (3), (4) and (5)
$W = m\dfrac{v}{{t'}}\left( {\dfrac{1}{2}\dfrac{v}{{t'}}{t^2}} \right)$
$W = \dfrac{m}{2}\dfrac{{{v^2}}}{{t{'^2}}}{t^2}$
$\therefore$ The correct answer is $W = \dfrac{m}{2}\dfrac{{{v^2}}}{{t{'^2}}}{t^2}$

Note: It is also important to note here that we have only assumed linear motion and not any kind of circular motion, if we are given a Force V/s Displacement curve, work done can be calculated as the area under the curve.