Question

A light and a heavy object have the same momentum. What is the ratio of their kinetic energy?

Answer 1

Step 1: Given,

Let there be two bodies A and B, such that

The mass of body A is more than the mass of body B as $m_A>m_B$

The momentum of body A = momentum of body B

Step 2: Formula used,

$$\begin{gathered} \mathrm{Momentum}\left(\mathrm{p}\right)=\mathrm{mv} \\ \mathrm{Kinetic}\mathrm{energy}\left(\mathrm{K}\right)=\frac{1}{2}{\mathrm{mv}}^2 \end{gathered}$$

Step 3: Calculation,

$$\begin{gathered} \mathrm{as},\mathrm{K}=\frac{1}{2}{\mathrm{mv}}^2 \\ \mathrm{K}=\frac{{\left(\mathrm{mv}\right)}^2}{2\mathrm{m}} \\ \mathrm{K}=\frac{{\mathrm{p}}^2}{2\mathrm{m}} \\ \mathrm{As}\mathrm{momentum}\mathrm{is}\mathrm{constant},\mathrm{the}\mathrm{kinetic}\mathrm{energy}\mathrm{is}\mathrm{inversly}\mathrm{proportional}\mathrm{to}\mathrm{the}\mathrm{mass}. \\ \mathrm{So},\mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{kinetic}\mathrm{energy}\mathrm{of}\mathrm{body}\mathrm{A}\mathrm{to}\mathrm{body}\mathrm{B}. \\ \frac{{\mathrm{K}}_{\mathrm{A}}}{{\mathrm{K}}_{\mathrm{B}}}=\frac{{\mathrm{m}}_{\mathrm{B}}}{{\mathrm{m}}_{\mathrm{A}}}<1 \end{gathered}$$

Hence, the ratio of the kinetic energy of heavier objects to the lighter object in the case of constant momentum is less than 1.