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# A light and a heavy object have the same momentum. Find out the ratio of their kinetic energies. Which one has a larger kinetic energy?

Let a light object with mass $m$ is moving with $v$ velocity and a heavy object with mass $M$ is moving with velocity $V$.

It is given that they have the same momentum.

So, $mv=MV$

Or $\frac{v}{V}=\frac{M}{m}$ ......... $(i)$

Kinetic energy of light object, $K_m=\frac{1}{2}mv^2$

kinetic energy of heavy object, $K_M=\frac{1}{2}MV^2$

So, $\frac{K_m}{K_M}=\frac{\frac{1}{2}mv^2}{\frac{1}{2}MV^2}$

Or $\frac{K_m}{K_M}=\frac{mv^2}{MV^2}$

$=(\frac{m}{M})(\frac{M}{m})^2$

Or $\frac{K_m}{K_M}=\frac{M}{m}$

As we know, $M$>$m$

So, $K_m$>$K_M$

Therefore, the light object has larger kinetic energy.

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