Two bodies of masses `10 kg and 2 kg` are moving with velocities `(2 hati - 7 hatj + 3hat k) and (-10 hati + 35 hatj - 3hatk) m//s` respectively. Calculate the velocity of their centre of mass.

Options

A. 1. `2hati m s^(-1)`
B. 2. `2 hatk m s^(-1)`
C. 3. `(2hatj+2hatk) m s^(-1)`
D. 4. `(2hati+2hatj+2hatk) m s^(-1)`

Answer

To calculate the velocity of the center of mass (V_com) of the two bodies, we can use the following formula:

\[ V_{com} = \frac{m_1 V_1 + m_2 V_2}{m_1 + m_2} \]

Where:
- \(m_1\) and \(m_2\) are the masses of the two bodies.
- \(V_1\) and \(V_2\) are the velocities of the two bodies.

### Step 1: Identify the given values
- Mass of the first body, \(m_1 = 10 \, \text{kg}\)
- Velocity of the first body, \(V_1 = (2 \hat{i} - 7 \hat{j} + 3 \hat{k}) \, \text{m/s}\)
- Mass of the second body, \(m_2 = 2 \, \text{kg}\)
- Velocity of the second body, \(V_2 = (-10 \hat{i} + 35 \hat{j} - 3 \hat{k}) \, \text{m/s}\)

### Step 2: Substitute the values into the formula
\[ V_{com} = \frac{10 \cdot (2 \hat{i} - 7 \hat{j} + 3 \hat{k}) + 2 \cdot (-10 \hat{i} + 35 \hat{j} - 3 \hat{k})}{10 + 2} \]

### Step 3: Calculate the numerator
First, calculate \(m_1 V_1\) and \(m_2 V_2\):

\[ m_1 V_1 = 10 \cdot (2 \hat{i} - 7 \hat{j} + 3 \hat{k}) = (20 \hat{i} - 70 \hat{j} + 30 \hat{k}) \]

\[ m_2 V_2 = 2 \cdot (-10 \hat{i} + 35 \hat{j} - 3 \hat{k}) = (-20 \hat{i} + 70 \hat{j} - 6 \hat{k}) \]

Now, add these two results together:

\[ m_1 V_1 + m_2 V_2 = (20 \hat{i} - 70 \hat{j} + 30 \hat{k}) + (-20 \hat{i} + 70 \hat{j} - 6 \hat{k}) \]

Combining the components:

- For \(\hat{i}\): \(20 - 20 = 0\)
- For \(\hat{j}\): \(-70 + 70 = 0\)
- For \(\hat{k}\): \(30 - 6 = 24\)

So, we have:

\[ m_1 V_1 + m_2 V_2 = 0 \hat{i} + 0 \hat{j} + 24 \hat{k} = 24 \hat{k} \]

### Step 4: Calculate the denominator
The total mass is:

\[ m_1 + m_2 = 10 + 2 = 12 \, \text{kg} \]

### Step 5: Calculate the velocity of the center of mass
Now substitute back into the formula:

\[ V_{com} = \frac{24 \hat{k}}{12} = 2 \hat{k} \, \text{m/s} \]

### Final Answer
The velocity of the center of mass is:

\[ V_{com} = 2 \hat{k} \, \text{m/s} \]

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