Question

A light and a heavy object have the same momentum. Find out the ratio of their kinetic energies. Which one has a larger kinetic energy?

Answer 1

Step 1. Given data

The momentum of the light object and heavy object is the same.

Step 2. Formula used

1. Momentum $=mv$
2. Kinetic Energy$\left(K.E\right)$$=\frac{1}{2}m{v}^2$

Where $m$ is the mass and $v$ is the velocity of the object.

Step 3. Calculate the ratio of their kinetic energies

Let $m$ is the mass and $v$ is the velocity of the lighter object and $M$ is the mass and $V$ is the velocity of the heavy object.

Since both have the same momentum

$$\begin{gathered} \therefore mv=MV \\ \Rightarrow \frac{M}{m}=\frac{v}{V} \end{gathered}$$

Clearly $v>V$

The kinetic energy of the lighter object will be be

${\left(K.E\right)}_1=\frac{1}{2}m{v}^2$

The kinetic energy of the heavier object will be

${\left(K.E\right)}_2=\frac{1}{2}M{V}^2$

$$\begin{gathered} \therefore {\left(K.E\right)}_1:{\left(K.E\right)}_2 \\ =\frac{1}{2}m{v}^2:\frac{1}{2}M{V}^2 \\ =m{v}^2:\frac{mv}{V}{V}^2\left[\because \frac{m}{M}=\frac{V}{v}\right] \\ =v:V \end{gathered}$$

So the ratio of the kinetic energy is $v:V$.

Step 4. Find the larger kinetic energy

$\because v>V$

So ${\left(K.E\right)}_1>{\left(K.E\right)}_2$

Hence, the ratio of the kinetic energies is $v:V$ and the kinetic energy of the lighter object is greater than the kinetic energy of the heavier object.