Two bodies have kinetic energies in the ratio 4:1. What is the ratio of their momenta?

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Solution

Step 1: Given,
Ratio of kinetic energy K₁: K₂=4 :1
Step 2: Formula used,
Kinetic energy K and momentum p is related as $K = \frac{p^{2}}{2 m}$ where m is the mass.
Step 3: Finding the ratio of momenta
We can write as,
$\frac{K_{1}}{K_{2}} = \frac{p_{1}^{2}}{p_{2}^{2}} \frac{p_{1}}{p_{2}} = \sqrt{\frac{K_{1}}{K_{2}}} \frac{p_{1}}{p_{2}} = \sqrt{\frac{4}{1}} \frac{p_{1}}{p_{2}} = \frac{2}{1}$
Hence, the ratio of the momenta is $2 : 1$ .

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