Question

A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of 2:1. After disintegration they will move

• A. In opposite directions with speed in the ratio of 1:2 respectively
• B. In opposite directions with speed in the ratio of 2:1 respectively
• C. In the same direction with same speed.
• D. In opposite directions with the same speed.

Answer 1

The Correct Option is A

This problem concerns a nucleus that breaks into two smaller nuclei. The mass ratio of these resulting nuclei is 2:1. The objective is to determine the ratio of their speeds post-disintegration, utilizing the principle of conservation of momentum.

Concept: Conservation of Momentum

The law of conservation of momentum dictates that the total momentum of an isolated system remains constant. For a system where no external forces are present, the initial momentum equals the final momentum. In this scenario, the nucleus is initially stationary, thus possessing zero initial momentum.

Solution Steps

1. System Definition: The initial momentum of the nucleus is zero as it is at rest.
2. Assign Variables:
   • Let the masses of the two daughter nuclei be \( m_1 \) and \( m_2 \). Given the mass ratio of 2:1, we can set \( m_1 = 2m \) and \( m_2 = m \).
   • Let their respective velocities be \( v_1 \) and \( v_2 \).
3. Apply Momentum Conservation: The initial momentum (0) must equal the final momentum. Since the nuclei move in opposite directions, one velocity will be negative relative to the other. Thus: \( 0 = m_1 \cdot v_1 + m_2 \cdot (-v_2) \). Substituting the masses: \( 0 = (2m) \cdot v_1 - m \cdot v_2 \). This rearranges to \( 2m \cdot v_1 = m \cdot v_2 \).
4. Equation Simplification: Divide both sides by \( m \) to yield \( 2v_1 = v_2 \).
5. Speed Ratio Determination: From \( 2v_1 = v_2 \), we can express the ratio of speeds as \( v_1 : v_2 = 1 : 2 \).

Outcome

Consequently, following disintegration, the two smaller nuclei will move in opposite directions. Their speeds will be in the ratio of 1:2.

The precise answer is: "In opposite directions with speed in the ratio of 1:2 respectively".