AIIMS 2004 Physics Radioactive Decay MCQ Question

To solve the problem of a nucleus of mass number $A$ emitting an α-particle with speed $v$, we need to apply the principle of conservation of momentum. Let's break down the problem step-by-step.

Step 1: Understand the system

1. Initial state: The parent nucleus (mass number $A$) is at rest, so its initial momentum is: $p_{\text{initial}} = 0$

2. Final state: After emitting an α-particle (which has a mass number of 4), we have two particles:

   • The α-particle with mass number 4 and speed $v$.
   • The daughter nucleus with mass number $A - 4$ and an unknown speed $V$.

Step 2: Apply conservation of momentum

According to the law of conservation of momentum, the total momentum before the decay must equal the total momentum after the decay. Therefore, we can write:

$p_{\text{initial}} = p_{\text{α}} + p_{\text{daughter}}$

This can be expressed mathematically as: $0 = (4)(v) + (A - 4)(-V)$

Here, we take the direction of the α-particle's motion as positive, and thus the daughter's recoil velocity $V$ is negative.

Step 3: Rearranging the equation

Rearranging the above equation gives: $0 = 4v - (A - 4)V$ $(A - 4)V = 4v$ $V = \frac{4v}{A - 4}$

Step 4: Evaluate the answer

The speed of the daughter nucleus $V$ is given by: $V = \frac{4v}{A - 4}$

This matches option C, confirming that the correct answer is indeed $V = \frac{4v}{A - 4}$.

Step 5: Clarification of other options

• Option A: $\frac{2v}{A + 4}$: This expression does not correctly apply the conservation of momentum principles, as it suggests a relationship that doesn't account for the correct mass ratios.

• Option B: $\frac{4v}{A + 4}$: Similar to option A, this does not respect the conservation of momentum since it incorrectly adds the mass numbers, leading to an incorrect denominator.

• Option D: $\frac{2v}{A - 4}$: This is also incorrect, as it fails to account for the mass of the α-particle properly and reduces the numerator incorrectly.

Conclusion

The application of conservation of momentum has shown that the speed of the daughter nucleus after the emission of the α-particle is $\frac{4v}{A - 4}$. Thus, the correct option is C. This problem beautifully illustrates the principle of conservation of momentum in nuclear decay processes.