An object of mass 3m splits into three equal fragments. Two fragments have velocities `vhatj` and `v hati`. The velocity of the third fragment is

Options

A. `v(j-i)`
B. `v(i-j)`
C. `-v(i+j)`
D. `(v(i+j))/sqrt(2)`

Explanation

To solve the problem, we need to find the velocity of the third fragment after an object of mass \(3m\) splits into three equal fragments. Given that two fragments have velocities \(\hat{v}j\) and \(\hat{v}i\), we can use the principle of conservation of momentum.

### Step-by-Step Solution:

1. **Identify the Mass of Each Fragment**:
Since the total mass of the object is \(3m\) and it splits into three equal fragments, the mass of each fragment is:
\[ m = \frac{3m}{3} = m \]

2. **Initial Momentum**:
Initially, the object is at rest, so the initial momentum \(P_{initial}\) is:
\[ P_{initial} = 0 \]

3. **Final Momentum**:
After the split, the momentum of each fragment can be expressed as follows:
- For the first fragment (velocity \(\hat{v}j\)):
\[ P_1 = m \cdot \hat{v}j \]
- For the second fragment (velocity \(\hat{v}i\)):
\[ P_2 = m \cdot \hat{v}i \]
- For the third fragment (velocity \(\vec{v_3}\)):
\[ P_3 = m \cdot \vec{v_3} \]

4. **Apply Conservation of Momentum**:
According to the conservation of momentum:
\[ P_{initial} = P_1 + P_2 + P_3 \]
Substituting the values:
\[ 0 = m \cdot \hat{v}j + m \cdot \hat{v}i + m \cdot \vec{v_3} \]

5. **Simplify the Equation**:
Dividing the entire equation by \(m\) (since \(m \neq 0\)):
\[ 0 = \hat{v}j + \hat{v}i + \vec{v_3} \]

6. **Solve for \(\vec{v_3}\)**:
Rearranging the equation to solve for \(\vec{v_3}\):
\[ \vec{v_3} = -(\hat{v}i + \hat{v}j) \]

7. **Final Expression**:
Thus, the velocity of the third fragment is:
\[ \vec{v_3} = -\hat{v}i - \hat{v}j \]

### Conclusion:
The velocity of the third fragment is:
\[ \vec{v_3} = -\hat{v}i - \hat{v}j \]