Two bodies are having kinetic energies in the ratio 16 : 9 . If they have same linear momentum, the ratio of…

The kinetic energy and the momentum of a moving object are related by

$K=\frac{p^2}{2m} ...\left(1\right)$

Hence, the ratio of the kinetic energies of two objects having masses $m$ and $m\text{'}$ can be calculated as follows:

$\begin{array}{rcl}\frac{K}{K^{\text{'}}}& =& \frac{{\displaystyle \frac{p^2}{2m}}}{{\displaystyle \frac{p^2}{2m\text{'}}}}\\ & =& \frac{m\text{'}}{m} ...\left(2\right)\end{array}$

Substitute the ratio of the kinetic energies into equation (2) and simplify to obtain the ratio of their masses.

$\begin{array}{rcl}& \Rightarrow & \frac{16}{9}=\frac{m\text{'}}{m}\\ & \Rightarrow & m:m\text{'}=9:16\end{array}$