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 will analyze the momentum of both balls A and B before and after the collision, and then evaluate the given statements based on our calculations.

### Step 1: Identify Initial Momenta
- For ball A, the initial momentum \( P_{A_i} = 2\hat{i} + 4\hat{j} \).
- For ball B, the initial momentum \( P_{B_i} = 6\hat{j} \).

### Step 2: Identify Final Momentum of Ball B
- After the collision, the momentum of ball B is given as \( P_{B_f} = 10\hat{j} \).

### Step 3: Calculate Change in Momentum for Ball B
- The change in momentum for ball B can be calculated as:
\[ \Delta P_B = P_{B_f} - P_{B_i} = 10\hat{j} - 6\hat{j} = 4\hat{j} \]
- The magnitude of this change is:
\[ |\Delta P_B| = 4 \]

### Step 4: Apply Conservation of Momentum
According to the law of conservation of momentum, the total momentum before the collision should equal the total momentum after the collision. Thus, we have:
\[ P_{A_i} + P_{B_i} = P_{A_f} + P_{B_f} \]
Substituting the known values:
\[ (2\hat{i} + 4\hat{j}) + (6\hat{j}) = P_{A_f} + (10\hat{j}) \]
This simplifies to:
\[ 2\hat{i} + 10\hat{j} = P_{A_f} + 10\hat{j} \]

### Step 5: Solve for Final Momentum of Ball A
From the equation above, we can isolate \( P_{A_f} \):
\[ P_{A_f} = 2\hat{i} + 10\hat{j} - 10\hat{j} = 2\hat{i} \]

### Step 6: Calculate Change in Momentum for Ball A
Now we can calculate the change in momentum for ball A:
\[ \Delta P_A = P_{A_f} - P_{A_i} = (2\hat{i}) - (2\hat{i} + 4\hat{j}) = -4\hat{j} \]
The magnitude of the change is:
\[ |\Delta P_A| = 4 \]

### Step 7: Evaluate the Statements
Now, we evaluate the statements provided in the question:
1. **Statement A**: After collision, momentum of A is \( 4\hat{i} \) - **Incorrect** (it is \( 2\hat{i} \)).
2. **Statement B**: After collision, momentum of A is parallel to momentum of B - **Incorrect** (they are perpendicular).
3. **Statement C**: After collision, momentum of A is \( 2\hat{i} \) - **Correct**.
4. **Statement D**: \( |\Delta P_A + \Delta P_B| = 0 \) - **Incorrect** (the direction matters).

### Conclusion
The only correct statement is **C**: After collision, momentum of A is \( 2\hat{i} \).

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