Two bodies of mases 5kg and 3kg are moving towards each other with `2ms^(-1)` and `4ms^(-1)` respectively. Then velocity of centre of mass is

Options

A. `0.25ms^(-1)` towards `3kg`
B. `0.5ms^(-1)` towards `5kg`
C. `0.25ms^(-1)` towards `5kg`
D. `0.5ms^(-1)` towards `3kg`

Answer

To solve the problem of finding 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**:
- Let \( m_1 = 5 \, \text{kg} \) (mass of the first body)
- Let \( m_2 = 3 \, \text{kg} \) (mass of the second body)
- Let \( v_1 = 2 \, \text{m/s} \) (velocity of the first body, moving to the right)
- Let \( v_2 = -4 \, \text{m/s} \) (velocity of the second body, moving to the left; we take left as negative)

2. **Use the formula for the velocity of the center of mass**:
The velocity of the center of mass (\( V_{cm} \)) is given by:
\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

3. **Substitute the values into the formula**:
\[ V_{cm} = \frac{(5 \, \text{kg} \times 2 \, \text{m/s}) + (3 \, \text{kg} \times -4 \, \text{m/s})}{5 \, \text{kg} + 3 \, \text{kg}} \]

4. **Calculate the numerator**:
\[ V_{cm} = \frac{(10 \, \text{kg m/s}) + (-12 \, \text{kg m/s})}{8 \, \text{kg}} \]
\[ V_{cm} = \frac{-2 \, \text{kg m/s}}{8 \, \text{kg}} \]

5. **Simplify the expression**:
\[ V_{cm} = -\frac{1}{4} \, \text{m/s} \quad \text{or} \quad V_{cm} = -0.25 \, \text{m/s} \]

6. **Interpret the result**:
The negative sign indicates that the center of mass is moving in the direction of the mass with the larger magnitude of velocity (which is the 5 kg mass moving at 2 m/s).

### Final Answer:
The velocity of the center of mass is \( -0.25 \, \text{m/s} \), which means it is moving towards the 5 kg mass.

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