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, we will apply the principle of conservation of momentum. The conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant.

### Step-by-Step Solution:

1. **Identify Initial and Final States**:
- Initially, we have a nucleus of mass \( m \) moving with velocity \( \bar{v} \).
- After emitting an alpha particle, we have two objects: the alpha particle with mass \( m_1 \) and velocity \( \bar{v}_1 \), and the remaining nucleus with mass \( m_2 \) and velocity \( \bar{v}_2 \).

2. **Write the Conservation of Momentum Equation**:
- According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
- Therefore, we can write:
\[ m \bar{v} = m_1 \bar{v}_1 + m_2 \bar{v}_2 \]

3. **Analyze the Directions of Velocities**:
- The velocities \( \bar{v} \), \( \bar{v}_1 \), and \( \bar{v}_2 \) can be in different directions. We need to consider their vector nature.
- Since momentum is a vector quantity, we can break down the equation into components if necessary, but for simplicity, we will analyze the overall vector equation.

4. **Determine Relationships Between Velocities**:
- From the equation \( m \bar{v} = m_1 \bar{v}_1 + m_2 \bar{v}_2 \), we can infer that the resultant momentum vector \( m_1 \bar{v}_1 + m_2 \bar{v}_2 \) must be in the same direction as \( m \bar{v} \) for momentum to be conserved.
- This implies that the vector sum of \( m_1 \bar{v}_1 \) and \( m_2 \bar{v}_2 \) must equal the initial momentum vector \( m \bar{v} \).

5. **Evaluate the Given Options**:
- We need to check which of the options regarding the relationships between \( \bar{v} \), \( \bar{v}_1 \), and \( \bar{v}_2 \) is correct.
- The options suggest conditions about whether these velocities must be parallel or not.

6. **Conclusion**:
- The correct conclusion is that the vector sum \( m_1 \bar{v}_1 + m_2 \bar{v}_2 \) must be parallel to \( m \bar{v} \). Therefore, the correct option is:
\[ m_1 \bar{v}_1 + m_2 \bar{v}_2 \text{ must be parallel to } m \bar{v} \]

### Final Answer:
The correct option is **D**: \( m_1 \bar{v}_1 + m_2 \bar{v}_2 \text{ must be parallel to } m \bar{v} \).