STATEMENT-1: A body is projected from the ground with a velocity v at an angle with the horizontal direction, it reaches a maximum height (H) and reaches the ground after a time T = 2u sin `theta//g`
because STATEMENT-2: The vertical and horizontal motions can be treated independently

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 the two statements provided and verify their validity step by step.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- A body is projected from the ground with an initial velocity \( v \) at an angle \( \theta \) with the horizontal. The motion can be broken down into horizontal and vertical components.

2. **Breaking Down the Velocity**:
- The initial velocity \( v \) can be decomposed into two components:
- Vertical component: \( v_y = v \sin \theta \)
- Horizontal component: \( v_x = v \cos \theta \)

3. **Calculating Time to Reach Maximum Height**:
- At maximum height, the vertical component of the velocity becomes zero. Using the equation of motion:
\[ v_y = u_y + at \]
where \( v_y = 0 \), \( u_y = v \sin \theta \), and \( a = -g \) (acceleration due to gravity).
- Setting the equation:
\[ 0 = v \sin \theta - g t_{up} \]
- Rearranging gives:
\[ t_{up} = \frac{v \sin \theta}{g} \]

4. **Calculating Total Time of Flight**:
- The total time of flight \( T \) is twice the time taken to reach maximum height (since the time to go up is equal to the time to come down):
\[ T = 2 t_{up} = 2 \left(\frac{v \sin \theta}{g}\right) = \frac{2v \sin \theta}{g} \]
- This confirms Statement 1: The time of flight \( T = \frac{2v \sin \theta}{g} \).

5. **Independence of Vertical and Horizontal Motions**:
- The second statement asserts that vertical and horizontal motions can be treated independently. This is true because:
- The horizontal motion experiences no acceleration (assuming no air resistance), while the vertical motion is influenced only by gravity.
- Therefore, the horizontal and vertical motions can be analyzed separately.

6. **Conclusion**:
- Both Statement 1 and Statement 2 are true. Statement 2 correctly explains Statement 1, as the independence of the motions allows us to derive the time of flight.

### Final Answer:
Both statements are true. Statement 1 is correct, and Statement 2 provides the correct explanation for Statement 1.

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