STATEMENT-1 `:` A stone of mass m kg is dropped vertically down for the window of a train which is at rest and than the same stone is dropped from the window of the train accelerating with a `m//s^(2)`. The net force in the space acting on the stone is mg downward in both cases.
because
STATEMENT-2`:` Vertical and horizontal motions are independent of each other in two dimensional projectile motion.

Options

A. Statement-1 is True , Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True , Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True.

Answer

To solve the problem, we need to analyze both statements and determine their validity and relationship.

### Step-by-Step Solution:

1. **Understanding Statement 1**:
- A stone of mass \( m \) is dropped from a train that is at rest. The only force acting on the stone is its weight, which is \( mg \) downward.
- When the stone is dropped from a train that is accelerating with \( a \, \text{m/s}^2 \), the stone also experiences a downward force of \( mg \) when it is released. The stone will have an initial horizontal velocity equal to that of the train at the moment of release.

2. **Force Analysis**:
- In both cases (the train at rest and the train accelerating), the net force acting on the stone after it is dropped is \( mg \) downward.
- The acceleration of the stone after it is released is due only to gravity, which is \( g \), and this is independent of the train's acceleration.

3. **Understanding Statement 2**:
- In two-dimensional projectile motion, the vertical and horizontal motions are independent of each other.
- The vertical motion is influenced by gravity (an acceleration of \( g \)), while the horizontal motion does not experience any acceleration (assuming no air resistance).

4. **Conclusion**:
- Statement 1 is true because the net force acting on the stone is \( mg \) downward in both scenarios.
- Statement 2 is also true as it correctly describes the independence of vertical and horizontal motions in projectile motion.
- Therefore, Statement 2 serves as a correct explanation for Statement 1.

5. **Final Answer**:
- Both statements are true, and Statement 2 is the correct explanation for Statement 1. Thus, the answer is A.