AIPMT PRELIMS 2004 Physics Momentum and Energy MCQ Question

To understand the question, let's first define the concepts of momentum and kinetic energy.

Concepts

1. Momentum: The momentum $p$ of a particle is given by the equation: $p = mv$ where $m$ is the mass of the particle and $v$ is its velocity.

2. Kinetic Energy: The kinetic energy $E$ of a particle is given by: $E = \frac{1}{2} mv^2$

Given Information

• We have two particles:

  • Particle 1: mass $m_1$, velocity $v_1$, momentum $p_1 = m_1 v_1$, kinetic energy $E_1 = \frac{1}{2} m_1 v_1^2$
  • Particle 2: mass $m_2$, velocity $v_2$, momentum $p_2 = m_2 v_2$, kinetic energy $E_2 = \frac{1}{2} m_2 v_2^2$

• The question states that both particles have the same momentum: $p_1 = p_2$ This implies: $m_1 v_1 = m_2 v_2$

From this equation, we can express one velocity in terms of the other: $v_2 = \frac{m_1}{m_2} v_1$

Kinetic Energy Comparison

Now, we can substitute $v_2$ back into the kinetic energy formula for particle 2: $E_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_2 \left(\frac{m_1}{m_2} v_1\right)^2 = \frac{1}{2} m_2 \cdot \frac{m_1^2}{m_2^2} v_1^2 = \frac{1}{2} \frac{m_1^2}{m_2} v_1^2$

Now we can rewrite the kinetic energy of particle 1: $E_1 = \frac{1}{2} m_1 v_1^2$

Ratio of Kinetic Energies

To find the ratio of $E_1$ to $E_2$: $\frac{E_1}{E_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} \frac{m_1^2}{m_2} v_1^2} = \frac{m_1}{\frac{m_1^2}{m_2}} = \frac{m_2}{m_1}$

Since it is given that $m_1 > m_2$, we have: $\frac{m_2}{m_1} < 1$

This indicates that: $E_1 < E_2$

Conclusion

Therefore, the correct answer is D) $E_1 < E_2$.

Clarification of Other Options

• Option A: $\frac{E_1}{E_2} = \frac{m_1}{m_2}$: This is incorrect because we derived that $\frac{E_1}{E_2} = \frac{m_2}{m_1}$, not $\frac{m_1}{m_2}$.

• Option B: $E_1 > E_2$: This contradicts our finding that $E_1 < E_2$.

• Option C: $E_1 = E_2$: This is also incorrect as we established that $E_1$ and $E_2$ are not equal due to the different masses and the derived relationship.

Summary

In summary, given that the two particles have the same momentum, the particle with the larger mass (particle 1) has a smaller kinetic energy than the particle with the smaller mass (particle 2). Thus, the correct answer is D) $E_1 < E_2$.