Statement-1: Two particles of mass 1 kg and 3 kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is `2 m//s`, their centre of mass has a velocity of `0.5 m//s`. When the relative velocity of approach becomes `3 m//s` the velocity of the centre of mass is `0.75 m//s`.
Statement-2: The total kinetic energy as seen from ground is `1/2muv_(rel)^(2)+1/2mv_(c)^(2)` and in absence of external force, total energy remains conserved.

Options

A. Only statement1 is correct
B. Only statement2 is correct
C. `10 ms ^(-1)`
D. `1.25 ms ^(-1)`

Answer

To solve the problem, we will analyze both statements provided in the question and determine their validity based on the principles of physics.

### Step-by-Step Solution:

1. **Understanding the System**:
We have two particles with masses \( m_1 = 1 \, \text{kg} \) and \( m_2 = 3 \, \text{kg} \) moving towards each other under their mutual gravitational attraction. There are no external forces acting on them.

**Hint**: Identify the system and the forces acting on it.

2. **Conservation of Momentum**:
Since there are no external forces acting on the system, the total momentum of the system is conserved. This implies that the velocity of the center of mass (COM) of the system should remain constant over time.

**Hint**: Recall that the momentum of a closed system is conserved.

3. **Analyzing Statement 1**:
- Initially, the relative velocity of approach is \( 2 \, \text{m/s} \) and the velocity of the center of mass is \( 0.5 \, \text{m/s} \).
- Later, when the relative velocity of approach becomes \( 3 \, \text{m/s} \), the velocity of the center of mass changes to \( 0.75 \, \text{m/s} \).
- Since the velocity of the center of mass is changing, this contradicts the principle of conservation of momentum, which states that the velocity of the center of mass should remain constant in the absence of external forces.

**Conclusion for Statement 1**: This statement is **false**.

**Hint**: Check if the velocity of the center of mass remains constant when no external forces are acting.

4. **Analyzing Statement 2**:
- The second statement discusses the total kinetic energy of the system, given by the formula:
\[ KE = \frac{1}{2} m u v_{\text{rel}}^2 + \frac{1}{2} m v_c^2 \]
- Here, \( v_{\text{rel}} \) is the relative velocity between the two particles, and \( v_c \) is the velocity of the center of mass.
- It states that in the absence of external forces, the total energy remains conserved. This is a true statement as energy conservation is a fundamental principle in physics.

**Conclusion for Statement 2**: This statement is **true**.

**Hint**: Remember the principle of conservation of energy and how it applies to isolated systems.

5. **Final Conclusion**:
Given that Statement 1 is false and Statement 2 is true, the correct option based on the analysis is that Statement 1 is incorrect, and Statement 2 is correct.

### Final Answer:
- Statement 1: False
- Statement 2: True