Three bodies having masses 5 kg, 4 kg and 2 kg is moving at the speed of 5 `m//s and 2 `m//s and 5 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 use the formula for the velocity of the center of mass (V_cm):

\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2 + m_3 v_3}{m_1 + m_2 + m_3} \]

Where:
- \(m_1, m_2, m_3\) are the masses of the bodies,
- \(v_1, v_2, v_3\) are the velocities of the bodies.

### Step 1: Identify the masses and velocities
- Masses:
- \(m_1 = 5 \, \text{kg}\)
- \(m_2 = 4 \, \text{kg}\)
- \(m_3 = 2 \, \text{kg}\)

- Velocities:
- \(v_1 = 5 \, \text{m/s}\)
- \(v_2 = 2 \, \text{m/s}\)
- \(v_3 = 5 \, \text{m/s}\)

### Step 2: Calculate the total momentum
Calculate the total momentum of the system:

\[ \text{Total Momentum} = m_1 v_1 + m_2 v_2 + m_3 v_3 \]

Substituting the values:

\[ = (5 \, \text{kg} \times 5 \, \text{m/s}) + (4 \, \text{kg} \times 2 \, \text{m/s}) + (2 \, \text{kg} \times 5 \, \text{m/s}) \]

Calculating each term:

\[ = 25 \, \text{kg m/s} + 8 \, \text{kg m/s} + 10 \, \text{kg m/s} \]

Adding these together:

\[ = 25 + 8 + 10 = 43 \, \text{kg m/s} \]

### Step 3: Calculate the total mass
Calculate the total mass of the system:

\[ \text{Total Mass} = m_1 + m_2 + m_3 \]

Substituting the values:

\[ = 5 \, \text{kg} + 4 \, \text{kg} + 2 \, \text{kg} = 11 \, \text{kg} \]

### Step 4: Calculate the velocity of the center of mass
Now, substitute the total momentum and total mass into the center of mass velocity formula:

\[ V_{cm} = \frac{43 \, \text{kg m/s}}{11 \, \text{kg}} \]

Calculating this gives:

\[ V_{cm} \approx 3.91 \, \text{m/s} \]

### Step 5: Round to the appropriate significant figures
Rounding to two decimal places, we find:

\[ V_{cm} \approx 3.91 \, \text{m/s} \]

### Final Answer
The magnitude of the velocity of the center of mass is approximately **3.91 m/s**.

---