Two objects of masses `200g` and `500g` have velocities of `10i m//s` and `(3i + 5j) m//s` respectively. The velocity of their centre of mass is

Options

A. `5i-25j`
B. `5/7 i-25j`
C. `5i+25/7j`
D. `25i-5/7j `

Explanation

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

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

Where:
- \( m_1 \) and \( m_2 \) are the masses of the objects,
- \( V_1 \) and \( V_2 \) are the velocity vectors of the objects.

### Step 1: Identify the given values
- Mass of the first object, \( m_1 = 200 \, \text{g} = 0.2 \, \text{kg} \) (convert grams to kilograms)
- Velocity of the first object, \( V_1 = 10 \hat{i} \, \text{m/s} \)
- Mass of the second object, \( m_2 = 500 \, \text{g} = 0.5 \, \text{kg} \) (convert grams to kilograms)
- Velocity of the second object, \( V_2 = 3 \hat{i} + 5 \hat{j} \, \text{m/s} \)

### Step 2: Substitute the values into the formula
Substituting the values into the center of mass velocity formula:

\[ V_{cm} = \frac{(0.2 \, \text{kg} \cdot 10 \hat{i}) + (0.5 \, \text{kg} \cdot (3 \hat{i} + 5 \hat{j}))}{0.2 \, \text{kg} + 0.5 \, \text{kg}} \]

### Step 3: Calculate the numerator
Calculating the momentum contributions from each mass:

- For the first mass:
\[ 0.2 \cdot 10 \hat{i} = 2 \hat{i} \]

- For the second mass:
\[ 0.5 \cdot (3 \hat{i} + 5 \hat{j}) = 1.5 \hat{i} + 2.5 \hat{j} \]

Now, add these two results together:
\[ 2 \hat{i} + (1.5 \hat{i} + 2.5 \hat{j}) = (2 + 1.5) \hat{i} + 2.5 \hat{j} = 3.5 \hat{i} + 2.5 \hat{j} \]

### Step 4: Calculate the denominator
Now, calculate the total mass:
\[ 0.2 + 0.5 = 0.7 \, \text{kg} \]

### Step 5: Calculate the velocity of the center of mass
Now, substitute the results back into the formula:
\[ V_{cm} = \frac{3.5 \hat{i} + 2.5 \hat{j}}{0.7} \]

Dividing each component by \( 0.7 \):
\[ V_{cm} = \left(\frac{3.5}{0.7} \hat{i} + \frac{2.5}{0.7} \hat{j}\right) = 5 \hat{i} + \frac{25}{7} \hat{j} \]

### Final Result
Thus, the velocity of the center of mass is:
\[ V_{cm} = 5 \hat{i} + \frac{25}{7} \hat{j} \, \text{m/s} \]