Three bodies having masses 5 kg, 4 kg and 2 kg is moving at the speed of 5 `m//s and 2 `m//s respectively along X-axis.The magnitude of velocity of centre of mass is

Options

A. `1.0 m//s`
B. `4 m//s`
C. `0.9 m//s`
D. `1.3 m//s`

Answer

To find the magnitude of the velocity of the center of mass of the three bodies, we can follow these steps:

### Step 1: Identify the masses and their velocities
We have three bodies with the following masses and velocities:
- Mass \( m_1 = 5 \, \text{kg} \) with velocity \( v_1 = 5 \, \text{m/s} \)
- Mass \( m_2 = 4 \, \text{kg} \) with velocity \( v_2 = 2 \, \text{m/s} \)
- Mass \( m_3 = 2 \, \text{kg} \) with velocity \( v_3 = 5.5 \, \text{m/s} \)

### Step 2: Calculate the total mass
The total mass \( M \) of the system is the sum of the individual masses:
\[ M = m_1 + m_2 + m_3 = 5 \, \text{kg} + 4 \, \text{kg} + 2 \, \text{kg} = 11 \, \text{kg} \]

### Step 3: Calculate the total momentum
The total momentum \( P \) of the system is given by the sum of the momenta of each mass:
\[ P = m_1 v_1 + m_2 v_2 + m_3 v_3 \]
Substituting the values:
\[ P = (5 \, \text{kg} \times 5 \, \text{m/s}) + (4 \, \text{kg} \times 2 \, \text{m/s}) + (2 \, \text{kg} \times 5.5 \, \text{m/s}) \]
Calculating each term:
\[ P = 25 \, \text{kg m/s} + 8 \, \text{kg m/s} + 11 \, \text{kg m/s} = 44 \, \text{kg m/s} \]

### Step 4: Calculate the velocity of the center of mass
The velocity of the center of mass \( V_{cm} \) is given by the total momentum divided by the total mass:
\[ V_{cm} = \frac{P}{M} = \frac{44 \, \text{kg m/s}}{11 \, \text{kg}} = 4 \, \text{m/s} \]

### Final Answer
The magnitude of the velocity of the center of mass is \( 4 \, \text{m/s} \).

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