Two bodies of masses $m_{1}$ and $m_{2}$ have equal kinetic energies. What is the ratio of their linear momenta?

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Solution

Step 1: Data provided,
Mass of body 1 = $m_{1}$
Mass of body 2 = $m_{2}$
The kinetic energy of 1st body= kinetic energy of 2nd body = K (say)
Step 2: Formula used,
$P = m v, a n d K = \frac{1}{2} m v^{2}$
Where, P= momentum, m= mass, v = velocity, and K = kinetic energy
Step 2: Calculation,
$K = \frac{1}{2} m v^{2} \Rightarrow 2 K = \frac{{(m v)}^{2}}{m} \Rightarrow 2 K m = P^{2} \Rightarrow P = \sqrt{2 K m} S o, \frac{P_{1}}{P_{2}} = \frac{\sqrt{2 K m_{1}}}{\sqrt{2 K m_{2}}} = \frac{\sqrt{m_{1}}}{\sqrt{m_{2}}} O r, \frac{P_{1}}{P_{2}} = \frac{\sqrt{m_{1}}}{\sqrt{m_{2}}}$
Hence, the ratio of the momenta of the two bodies is, $\frac{\sqrt{m_{1}}}{\sqrt{m_{2}}}$ .

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