Two bodies of different masses `2kg` and `4kg` are moving with velocities `2m//s` and `10m//s` towards each other due to mutual gravitational attraction. Then the velocity of the centre of mass is

Options

A. 5`m/s`
B. 6`m/s`
C. Zero
D. 8`m/s`

Answer

To find the velocity of the center of mass of the two bodies, we can use the formula for the velocity of the center of mass (V_cm):

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

Where:
- \( m_1 \) is the mass of the first body,
- \( v_1 \) is the velocity of the first body,
- \( m_2 \) is the mass of the second body,
- \( v_2 \) is the velocity of the second body.

### Step 1: Identify the masses and velocities
- Let \( m_1 = 4 \, \text{kg} \) (mass of the first body)
- Let \( v_1 = 10 \, \text{m/s} \) (velocity of the first body, moving in the positive direction)
- Let \( m_2 = 2 \, \text{kg} \) (mass of the second body)
- Let \( v_2 = -2 \, \text{m/s} \) (velocity of the second body, moving in the opposite direction, hence negative)

### Step 2: Substitute the values into the formula
Now, substituting the values into the center of mass formula:

\[ V_{cm} = \frac{(4 \, \text{kg} \cdot 10 \, \text{m/s}) + (2 \, \text{kg} \cdot -2 \, \text{m/s})}{4 \, \text{kg} + 2 \, \text{kg}} \]

### Step 3: Calculate the numerator
Calculating the numerator:

\[ = (40 \, \text{kg m/s}) + (-4 \, \text{kg m/s}) = 40 - 4 = 36 \, \text{kg m/s} \]

### Step 4: Calculate the denominator
Calculating the denominator:

\[ = 4 \, \text{kg} + 2 \, \text{kg} = 6 \, \text{kg} \]

### Step 5: Calculate the velocity of the center of mass
Now, substituting the results back into the formula:

\[ V_{cm} = \frac{36 \, \text{kg m/s}}{6 \, \text{kg}} = 6 \, \text{m/s} \]

### Conclusion
The velocity of the center of mass is \( 6 \, \text{m/s} \) in the direction of the 4 kg mass.

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