If two particles of masses `3kg` and `6kg` which are at rest are separated by a distance of `15 m`. The two particles are moving towards each other under a mutual force of attraction. Then the ratio of distances travelled by the particles before collision is

Options

A. `2:1`
B. `1:2`
C. `1:3`
D. `3:1`

Explanation

To solve the problem of finding the ratio of distances travelled by two particles of masses 3 kg and 6 kg before they collide, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Masses and Initial Distance**:
- Let \( m_1 = 3 \, \text{kg} \) (mass of the first particle)
- Let \( m_2 = 6 \, \text{kg} \) (mass of the second particle)
- The initial distance between the two particles is \( d = 15 \, \text{m} \).

2. **Understand the Concept of Center of Mass**:
- Since the particles are moving under mutual attraction, the center of mass of the system will remain stationary.
- The position of the center of mass \( x_{cm} \) can be expressed as:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- Here \( x_1 \) and \( x_2 \) are the positions of the two masses.

3. **Set Up the Equation for Center of Mass**:
- Let \( R_1 \) be the distance travelled by the first particle (3 kg) and \( R_2 \) be the distance travelled by the second particle (6 kg).
- The center of mass will not change its position, so we can differentiate the center of mass equation:
\[ 0 = m_1 \, dR_1 + m_2 \, dR_2 \]
- This implies:
\[ m_1 R_1 + m_2 R_2 = 0 \]

4. **Relate the Distances**:
- Rearranging gives:
\[ m_1 R_1 = -m_2 R_2 \]
- Since the distances are in opposite directions, we can drop the negative sign:
\[ m_1 R_1 = m_2 R_2 \]

5. **Substitute the Masses**:
- Substitute \( m_1 = 3 \, \text{kg} \) and \( m_2 = 6 \, \text{kg} \):
\[ 3 R_1 = 6 R_2 \]

6. **Find the Ratio of Distances**:
- Dividing both sides by \( R_2 \) gives:
\[ \frac{R_1}{R_2} = \frac{6}{3} = 2 \]
- Therefore, the ratio of distances travelled by the two particles before collision is:
\[ R_1 : R_2 = 2 : 1 \]

### Conclusion:
The ratio of distances travelled by the particles before collision is \( 2 : 1 \).