Question

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.

Answer 1

Let the masses of the two particles be m₁ and m₂.
Given:
m₁ = 1 kg
m₂ = 4 kg

Now,
Kinetic energy of the first particle = Kinetic energy of the second particle

\[\left( \frac{1}{2} \right) m_1 v_1^2 = \left( \frac{1}{2} \right) m_2 v_2^2 \]

\[ \Rightarrow \frac{m_1}{m_2} = \frac{v_2^2}{v_1^2}\]

\[ \Rightarrow \frac{v_2}{v_1} = \sqrt{\frac{m_1}{m_2}}\]

\[ \Rightarrow \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}}\]

\[\text{ The ratio of linear momenta (mv) of the two particles,} \]

\[\frac{P_1}{P_2} = \frac{m_1 v_1}{m_2 v_2} = \frac{m_1}{m_2}\sqrt{\frac{m_2}{m_1}}\]

\[ = \sqrt{\frac{m_1}{m_2}} = \sqrt{\frac{1}{4}} = \frac{1}{2}\]

\[ \Rightarrow P_1 : P_2 = 1: 2\]

Therefore, the ratio of linear momenta is 1:2.