An object at rest in space suddenly explodes into three parts of same mass. The momentum of the two parts are `2phati` and `phatj`. The momentum of the third part

Options

A. will have a magnitude `psqrt3`
B. will have a magnitude `psqrt5`
C. will have a magnitude p
D. will have a magnitude 2p.

Explanation

To solve the problem, we will apply the principle of conservation of momentum. Since the object is initially at rest, its total initial momentum is zero. After the explosion, the total momentum of the three parts must also equal zero.

### Step-by-Step Solution:

1. **Identify the given momenta:**
- The momentum of the first part is \( \mathbf{p_1} = 2p \hat{i} \).
- The momentum of the second part is \( \mathbf{p_2} = p \hat{j} \).
- Let the momentum of the third part be \( \mathbf{p_3} \).

2. **Apply the conservation of momentum:**
- The total momentum before the explosion is zero (since the object is at rest).
- Therefore, the total momentum after the explosion must also be zero:
\[ \mathbf{p_1} + \mathbf{p_2} + \mathbf{p_3} = 0 \]

3. **Substitute the known momenta into the equation:**
\[ 2p \hat{i} + p \hat{j} + \mathbf{p_3} = 0 \]

4. **Rearranging the equation to solve for \( \mathbf{p_3} \):**
\[ \mathbf{p_3} = - (2p \hat{i} + p \hat{j}) \]
\[ \mathbf{p_3} = -2p \hat{i} - p \hat{j} \]

5. **Final expression for the momentum of the third part:**
- Thus, the momentum of the third part is:
\[ \mathbf{p_3} = -2p \hat{i} - p \hat{j} \]