Two objects of mass 4 kg and 2 kg have the same momentum. Find the ratio of their kinetic energies.

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Solution

Here, $m_{1} = 4 k g$ and $m_{2} = 2 k g$ .
Since both have the same momentum, $P_{1} = P_{2} \Rightarrow m_{1} v_{1} = m_{2} v_{2} \Rightarrow 4 v_{1} = 2 v_{2} \Rightarrow \frac{v_{1}}{v_{2}} = \frac{1}{2}$
Now, ratio of their kinetic energies, $\frac{K E_{1}}{K E_{2}} = \frac{\frac{1}{2} m_{1} {v_{1}}^{2}}{\frac{1}{2} m_{2} {v_{2}}^{2}} = \frac{\frac{1}{2} \times 4 {v_{1}}^{2}}{\frac{1}{2} \times 2 {v_{2}}^{2}} = \frac{2 {v_{1}}^{2}}{{v_{2}}^{2}} = \frac{1}{2}$

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