Since for a particle the kinetic energy is given by $\mathit{K}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}^{\mathbf{2}}$and the momentum by $\overrightarrow{\mathbf{P}}\mathbf{=}\mathit{m}\overrightarrow{\mathbf{v}}$$\mathit{K}\mathbf{=}{\mathit{p}}^{\mathbf{2}}\mathbf{/}\mathbf{2}\mathit{m}$, it is easy to show that . How, then, is it possible to have an event duringwhich the total momentum of the system is constant but the totalkinetic energy changes?

Kinetic Energy

$\mathit{K}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}^{\mathbf{2}}$

Momentum

$\overrightarrow{\mathbf{P}}\mathbf{=}\mathit{m}\overrightarrow{\mathbf{v}}$

The moment is a vector quantity,Supposethe magnitude of the momentum of two objects is the same, but the direction is different then their moment can be given as p and -p and the total moment is zero or constant.

But kinetic energy is a scalar quantity and$K=p^2/2m$ is always positive. Therefore, it is positive for both the momentums p and -p. So kinetic energy is not zero in this event it is changing.

Hence, we can conclude that since kinetic energy is a scalar quantity and momentum is a vector quantity therefore it is possible to have an event during which the total momentum of the system is constant but the total kinetic energy changes.