A nucleus moving with velocity `bar(v)` emits an `alpha`-particle. Let the velocities of the `alpha`-particle and the remaining nucleus be `bar(v)_(1)` and `bar(v)_(2)` and their masses be `m_(1)` and `(m_2)` then,

Options

A. `underset (v) rarr , underset (v_(1))rarr and underset (v_(2))rarr` must be parallel to each other
B. none of the two of `underset (v) rarr , underset (v_(1))rarr and underset (v_(2))rarr` should be paralle to each other
C. ` underset (v_(1))rarr + underset (v_(2))rarr` must be parallel to `underset (v) rarr `.
D. ` m_(1) underset (v_(1)) rarr + m_(2)underset (v_(2)) rarr ` must be parallel to `underset (v) rarr`

Explanation

To solve the problem of a nucleus emitting an alpha particle while moving with a certain velocity, we can apply the principle of conservation of momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Conditions
- We have a nucleus of mass \( m \) moving with velocity \( \bar{v} \).
- It emits an alpha particle of mass \( m_1 \) with velocity \( \bar{v}_1 \).
- The remaining nucleus has mass \( m_2 \) and moves with velocity \( \bar{v}_2 \).

### Step 2: Apply Conservation of Momentum
The law of conservation of momentum states that the total momentum before an event must equal the total momentum after the event if no external forces act on the system.

- **Initial Momentum**: The momentum of the moving nucleus before emitting the alpha particle is given by:
\[ p_{\text{initial}} = m \bar{v} \]

- **Final Momentum**: After the emission, the total momentum is the sum of the momenta of the alpha particle and the remaining nucleus:
\[ p_{\text{final}} = m_1 \bar{v}_1 + m_2 \bar{v}_2 \]

### Step 3: Set Initial Momentum Equal to Final Momentum
According to the conservation of momentum:
\[ m \bar{v} = m_1 \bar{v}_1 + m_2 \bar{v}_2 \]

### Step 4: Analyze the Direction of Velocities
- The velocities \( \bar{v}_1 \) and \( \bar{v}_2 \) can be in any direction relative to \( \bar{v} \). However, for the momentum equation to hold, the vector sum of the momenta must equal the initial momentum vector.

### Step 5: Conclusion
From the momentum conservation equation, we can conclude that the vector sum of the momenta of the alpha particle and the remaining nucleus must equal the momentum of the original nucleus. Therefore, the correct interpretation is that:
\[ m_1 \bar{v}_1 + m_2 \bar{v}_2 \text{ must be parallel to } m \bar{v} \]

Thus, the correct option is that the vector sum \( m_1 \bar{v}_1 + m_2 \bar{v}_2 \) must be parallel to \( m \bar{v} \).