Two bodies of `6kg` and `4kg` masses have their velocity `5hati - 2 hat j + 10 hat k` and `10hat i - 2 hat j + 5 hat k` respectively. Then, the velocity of their centre of mass is

Options

A. `5hati+2hatj-8hatk`
B. `7hati+2hatj-8hatk`
C. `7hati-2hatj+8hatk`
D. `5hati-2hatj+8hatk`

Answer

To find the velocity of the center of mass of two bodies with given masses and velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the masses and velocities:**
- Mass of body 1, \( m_1 = 6 \, \text{kg} \)
- Mass of body 2, \( m_2 = 4 \, \text{kg} \)
- Velocity of body 1, \( \vec{v_1} = 5 \hat{i} - 2 \hat{j} + 10 \hat{k} \)
- Velocity of body 2, \( \vec{v_2} = 10 \hat{i} - 2 \hat{j} + 5 \hat{k} \)

2. **Use the formula for the velocity of the center of mass:**
\[ \vec{v_{com}} = \frac{m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1 + m_2} \]

3. **Calculate the total mass:**
\[ m_1 + m_2 = 6 + 4 = 10 \, \text{kg} \]

4. **Calculate the x-component of the center of mass velocity:**
\[ v_{com_x} = \frac{m_1 v_{1x} + m_2 v_{2x}}{m_1 + m_2} \]
- Here, \( v_{1x} = 5 \) and \( v_{2x} = 10 \)
\[ v_{com_x} = \frac{6 \cdot 5 + 4 \cdot 10}{10} = \frac{30 + 40}{10} = \frac{70}{10} = 7 \, \hat{i} \]

5. **Calculate the y-component of the center of mass velocity:**
\[ v_{com_y} = \frac{m_1 v_{1y} + m_2 v_{2y}}{m_1 + m_2} \]
- Here, \( v_{1y} = -2 \) and \( v_{2y} = -2 \)
\[ v_{com_y} = \frac{6 \cdot (-2) + 4 \cdot (-2)}{10} = \frac{-12 - 8}{10} = \frac{-20}{10} = -2 \, \hat{j} \]

6. **Calculate the z-component of the center of mass velocity:**
\[ v_{com_z} = \frac{m_1 v_{1z} + m_2 v_{2z}}{m_1 + m_2} \]
- Here, \( v_{1z} = 10 \) and \( v_{2z} = 5 \)
\[ v_{com_z} = \frac{6 \cdot 10 + 4 \cdot 5}{10} = \frac{60 + 20}{10} = \frac{80}{10} = 8 \, \hat{k} \]

7. **Combine the components to find the total velocity of the center of mass:**
\[ \vec{v_{com}} = v_{com_x} + v_{com_y} + v_{com_z} = 7 \hat{i} - 2 \hat{j} + 8 \hat{k} \]

### Final Answer:
The velocity of the center of mass is:
\[ \vec{v_{com}} = 7 \hat{i} - 2 \hat{j} + 8 \hat{k} \]