A ball A moving with momentum `2hati + 4hatj` collides with identical ball B moving with momentum `6hatj` . After collision momentum of ball B is `10 hatj` . Which of the following statement is correct?

Options

A. After collision momentum of A is `4hati`
B. After collision momentum of A is parallel to momentum of B
C. After collision momentum of A is `2 hati`
D. If `vecP_A and vecP_B` are momenta of balls A and B, then `Delta(|vecP_A | + |vecP_B|) = 0`

Explanation

To solve the problem, we need to analyze the momentum of the two balls before and after the collision.

1. **Identify the initial momentum of both balls:**
- Ball A has an initial momentum of \( \vec{p_A} = 2\hat{i} + 4\hat{j} \).
- Ball B has an initial momentum of \( \vec{p_B} = 0\hat{i} + 6\hat{j} \).

2. **Calculate the total initial momentum:**
- The total initial momentum \( \vec{p_{initial}} \) is the vector sum of the momenta of both balls:
\[ \vec{p_{initial}} = \vec{p_A} + \vec{p_B} = (2\hat{i} + 4\hat{j}) + (0\hat{i} + 6\hat{j}) = 2\hat{i} + 10\hat{j} \]

3. **Identify the final momentum of ball B:**
- After the collision, the momentum of ball B is given as \( \vec{p_B'} = 0\hat{i} + 10\hat{j} \).

4. **Let’s denote the final momentum of ball A as \( \vec{p_A'} \).**
- We know that momentum is conserved, so the total final momentum \( \vec{p_{final}} \) must equal the total initial momentum:
\[ \vec{p_{final}} = \vec{p_A'} + \vec{p_B'} = \vec{p_A'} + (0\hat{i} + 10\hat{j}) \]

5. **Set up the equation for momentum conservation:**
\[ \vec{p_{initial}} = \vec{p_{final}} \implies 2\hat{i} + 10\hat{j} = \vec{p_A'} + (0\hat{i} + 10\hat{j}) \]

6. **Solve for \( \vec{p_A'} \):**
- By equating the components, we find:
- In the \( \hat{i} \) direction:
\[ 2 = p_{A_x}' \quad \text{(where \( p_{A_x}' \) is the \( \hat{i} \) component of \( \vec{p_A'} \))} \]
- In the \( \hat{j} \) direction:
\[ 10 = p_{A_y}' + 10 \implies p_{A_y}' = 0 \]

7. **Thus, the final momentum of ball A is:**
\[ \vec{p_A'} = 2\hat{i} + 0\hat{j} = 2\hat{i} \]

8. **Conclusion:**
- The correct statement is that after the collision, the momentum of ball A is \( 2\hat{i} \).