Two bodies of 6 kg and 4 kg masses have their velocity `5hati-2hatj+10hatk` and `10hati-2hatj+5hatk`, 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 (V_cm) of two bodies with given masses and velocities, we can follow these steps:

### Step 1: Identify the masses and velocities
- Mass of body 1 (m1) = 6 kg
- Velocity of body 1 (v1) = \(5 \hat{i} - 2 \hat{j} + 10 \hat{k}\) m/s
- Mass of body 2 (m2) = 4 kg
- Velocity of body 2 (v2) = \(10 \hat{i} - 2 \hat{j} + 5 \hat{k}\) m/s

### Step 2: Use the formula for the velocity of the center of mass
The formula for the velocity of the center of mass is given by:

\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

### Step 3: Calculate the total momentum in each direction
We will calculate the momentum in the x, y, and z directions separately.

#### For the x-direction:
\[ V_{cm_x} = \frac{m_1 v_{1x} + m_2 v_{2x}}{m_1 + m_2} \]
Where:
- \(v_{1x} = 5\) m/s
- \(v_{2x} = 10\) m/s

Calculating:
\[ V_{cm_x} = \frac{6 \times 5 + 4 \times 10}{6 + 4} = \frac{30 + 40}{10} = \frac{70}{10} = 7 \text{ m/s} \]

#### For the y-direction:
\[ V_{cm_y} = \frac{m_1 v_{1y} + m_2 v_{2y}}{m_1 + m_2} \]
Where:
- \(v_{1y} = -2\) m/s
- \(v_{2y} = -2\) m/s

Calculating:
\[ V_{cm_y} = \frac{6 \times (-2) + 4 \times (-2)}{6 + 4} = \frac{-12 - 8}{10} = \frac{-20}{10} = -2 \text{ m/s} \]

#### For the z-direction:
\[ V_{cm_z} = \frac{m_1 v_{1z} + m_2 v_{2z}}{m_1 + m_2} \]
Where:
- \(v_{1z} = 10\) m/s
- \(v_{2z} = 5\) m/s

Calculating:
\[ V_{cm_z} = \frac{6 \times 10 + 4 \times 5}{6 + 4} = \frac{60 + 20}{10} = \frac{80}{10} = 8 \text{ m/s} \]

### Step 4: Combine the components to find V_cm
Now we can combine the components to find the velocity of the center of mass:

\[ V_{cm} = V_{cm_x} \hat{i} + V_{cm_y} \hat{j} + V_{cm_z} \hat{k} \]
Substituting the values:
\[ V_{cm} = 7 \hat{i} - 2 \hat{j} + 8 \hat{k} \text{ m/s} \]

### Final Answer:
The velocity of the center of mass is:
\[ \boxed{7 \hat{i} - 2 \hat{j} + 8 \hat{k} \text{ m/s}} \]