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# The velocity of a body moving in a straight line is increased by applying a constant force F, for some distance in the direction of the motion. Prove that the increase in the kinetic energy of the body is equal to the work done by the force on the body.

Let a body with mass $m$ is moving with $u$ speed and a force $F$ is applied on the body and its velocity increases up to $v$. Let $a$ be the acceleration due to this force and it travels $s$ distance.

On using third equation of motion, $v^2=u^2+2as$

Or $s=\frac{v^2-u^2}{2a}$ ....... $(i)

Force applied $F=mass\times acceleration=ma$

So, work done $W=force\times displacement=Fs=(ma)s=mas$

On putting value of $s$ from $(i)$

$W=ma(\frac{v^2-u^2}{2a})$

Or $W=\frac{1}{2}mv^2-\frac{1}{2}mu^2$

Or work done$=$change in kinetic energy

Thus, it has been proved that the work done is equal to the change of kinetic energy.

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