A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of 2:1. After disintegration they will move
- In opposite directions with speed in the ratio of 1:2 respectively
- In opposite directions with speed in the ratio of 2:1 respectively
- In the same direction with same speed.
- In opposite directions with the same speed.
The Correct Option is A
Approach Solution - 1
The question involves the disintegration of a nucleus into two smaller nuclei with their mass ratio given as 2:1. We are to determine the speed ratio with which these nuclei move following disintegration. Let's solve this using principles of conservation of momentum.
Concept Involved: Conservation of Momentum
The principle of conservation of momentum states that if no external force acts on a system of particles, the total momentum of the system remains constant. In this case, the system is the disintegrating nucleus, which is initially at rest, so its initial momentum is zero.
Step-by-Step Solution
- Define the system: Initially, the momentum of the nucleus is zero because it is at rest.
- Assign masses and speeds:
- Let the masses of the two smaller nuclei be \( m_1 \) and \( m_2 \), with \( m_1 = 2m \) and \( m_2 = m \) (since the mass ratio is 2:1).
- Let their respective speeds be \( v_1 \) and \( v_2 \).
- Apply conservation of momentum: \(0 = m_1 \cdot v_1 + m_2 \cdot (-v_2)\)
- The negative sign for \( v_2 \) implies that the second nucleus moves in the opposite direction.
- This gives us the equation: \(2m \cdot v_1 = m \cdot v_2\).
- Simplify the equation:
- By canceling out the mass \( m \), we get: \(2v_1 = v_2\).
- Determine speed ratio:
- Rearranging, we find the speed ratio as \( v_1 : v_2 = 1 : 2 \).
Conclusion
Therefore, after disintegration, the two smaller nuclei will move in opposite directions with speeds in the ratio of 1:2.
The correct answer is: "In opposite directions with speed in the ratio of 1:2 respectively".
Approach Solution -2
Since the nucleus is at rest before disintegration, the total momentum of the system is zero. By the law of conservation of momentum, the momentum after disintegration must also be zero. The two smaller nuclei move in opposite directions, and the speed ratio is inversely proportional to their mass ratio.
Let the masses of the two nuclei be \( m_1 = 2m \) and \( m_2 = m \), and their speeds be \( v_1 \) and \( v_2 \), respectively. Using conservation of momentum:
\[ m_1 v_1 + m_2 v_2 = 0 \implies 2m \cdot v_1 + m \cdot v_2 = 0 \implies v_2 = -2v_1 \]
Thus, the speed ratio is \( 1 : 2 \), and they move in opposite directions.
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