The velocity of a body moving in a straight

Solution:

Consider a body or an object of mass $m$ moving with velocity $u$. Let a force $F$ be applied on the body so that the velocity attained by the body after travelling a distance $S$ is $v$ (Figure 5 ).

Work done by the force on the body is given by

$W=F S$ $\ldots$ (i)

Since velocity of the body changes so the body is accelerated. Let a be the acceleration of the body.

Therefore, according to Newton's second law of motion,

$F=m a$ ...(ii)

Using eqn. (ii) in eqn. (i), we get

$\mathrm{W}=(m a) \mathrm{S}$ $\ldots($ iii $)$

Now, using $v^{2}-u^{2}=2 a S$, we get

$\mathrm{S}=\frac{v^{2}-u^{2}}{2 a}$ $\ldots(i v)$

Using eqn. (iv) in eqn. (iii), we get

$\mathrm{W}=m a\left(\frac{v^{2}-u^{2}}{2 a}\right)=\frac{1}{2} m\left(v^{2}-u^{2}\right)$

or

$\mathrm{W}=\frac{1}{2} m v^{2}-\frac{1}{2} m u^{2}$ $\ldots(1)$

or

$W=$ Final K.E of body - Initial K.E of body

$=$ Change in K.E. of the body.

Thus, work done by a force on a body is equal to the change in kinetic energy of the body This is known as work-energy theorem.