A light and a heavy body have equal momentum. Which one has greater K.E.?

The lighter body

Let the two bodies be M and m such that M > m.

According to question : MV = mv ... where V and v are their respective velocities.

As M > m \(\Rightarrow\) V < v

Let the kinetic energies be : K and k ... where K is the kinetic energy of the heavier particle and k is kinetic energy of the lighter particle.

Let the momentum be : P 

We know : \(\text{Momentum} = \sqrt{2(\text{mass})(\text{kinetic energy})}\)

\(\Rightarrow \text{kinetic energy} = \frac{\text{momentum}^2}{2(\text{mass})}\)

\(\Rightarrow \text{kinetic energy} \propto \frac{1}{\text{mass}}\)

\(\Rightarrow \frac{K}{k} = {m}{M}\)

\(\text{Clearly, } K = \frac{m}{M}k \text{ and } k = \frac{M}{m} K\) 

\(\text{As } \frac{M}{m} > 1 \Rightarrow k > K\)