Find the ratio of the linear momenta of two particles of masses 1.0 kg and 4.0 kg if their kinetic energies are equal.

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Solution

Let the mass of the two particles be $m_{1}, m_{2},$ respectively.

$m_{1} = 1 k g, m_{2} = 4 k g$

$∴$ According to question,

$(\frac{1}{2}) m_{1} v_{1}^{2} = (\frac{1}{2}) m_{2} v_{2}^{2}$

$\Rightarrow \frac{m_{1}}{m_{2}} = \frac{v_{2}^{2}}{v_{1}^{2}}$

$\Rightarrow \frac{v_{2}}{v_{1}} = \sqrt{\frac{m_{1}}{m_{2}}}$

$\Rightarrow \frac{v_{1}}{v_{2}} = \sqrt{\frac{m_{2}}{m_{1}}}$

$N o w, \frac{m_{1} v_{1}}{m_{2} v_{2}} = \frac{m_{1}}{m_{2}} \sqrt{\frac{m_{2}}{m_{1}}}$

$= \sqrt{\frac{m_{1}}{m_{2}}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$

$\Rightarrow \frac{m_{1} v_{1}}{m_{2} v_{2}} = 1 : 2$

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