Question

Two bodies of masses 1 kg and 4 kg have equal linear momentum. What is the ratio of their kinetic energies?

Answer 1

Given data:

Mass of the first body is $m_1=1kg$

Mass of the second body is $m_2=4kg$

The momentum of both bodies is the same. Therefore, let the momentum of both bodies be $p$.

Relationship between kinetic energy and momentum:

Kinetic energy and momentum can be related as follows:

$KE=\frac{p^2}{2m}$

where $KE$ is kinetic energy, $p$ is momentum and $m$ is mass.

Obtaining the ratio of kinetic energies:

Let $K{E}_1$ be the kinetic energy of the first body and $K{E}_2$ be the kinetic energy of the second body. The ratio of both kinetic energies can be obtained as follows:

$$\begin{gathered} \frac{K{E}_1}{K{E}_2}=\frac{{\displaystyle \frac{p^2}{2m_1}}}{\frac{p^2}{2m_2}} \\ \frac{K{E}_1}{K{E}_2}=\frac{p^2}{2m_1}\times \frac{2m_2}{p^2} \\ \frac{K{E}_1}{K{E}_2}=\frac{m_2}{m_1} \\ \frac{K{E}_1}{K{E}_2}=\frac{4kg}{1kg} \\ \frac{K{E}_1}{K{E}_2}=\frac{4}{1} \end{gathered}$$

Therefore, the ratio of kinetic energies of the two bodies is $4:1$.