STATEMENT-1: A ball of mass m is released from the top of a smooth inclined plane. At the same time another ball of same mass allowed to fall freely from same height. When both balls reaches the bottom of the plane, momentum will not be equal
because
STATEMENT-2: Momentum depends on the direction of velocity as well as its magnitude.

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 determine the momentum of both balls when they reach the bottom of their respective paths.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two balls of mass \( m \).
- Ball A is released from the top of a smooth inclined plane.
- Ball B is allowed to fall freely from the same height \( h \).

2. **Determine the Velocity of Ball B (Free Fall)**:
- When Ball B falls freely, it starts from rest and falls a height \( h \).
- Using the equation of motion for free fall:
\[ v_B = \sqrt{2gh} \]
- The direction of Ball B's velocity when it reaches the bottom is vertically downward.

3. **Determine the Velocity of Ball A (Inclined Plane)**:
- Ball A also starts from rest and slides down the inclined plane.
- At the bottom of the incline, the potential energy at the top is converted to kinetic energy:
\[ mgh = \frac{1}{2} mv_A^2 \]
- Solving for \( v_A \):
\[ v_A = \sqrt{2gh} \]
- The direction of Ball A's velocity when it reaches the bottom is along the incline.

4. **Calculate the Momentum of Both Balls**:
- Momentum is given by the formula:
\[ p = mv \]
- For Ball B:
\[ p_B = m \cdot v_B = m \cdot \sqrt{2gh} \]
- For Ball A:
\[ p_A = m \cdot v_A = m \cdot \sqrt{2gh} \]

5. **Direction of Momentum**:
- The momentum of Ball B is directed downward.
- The momentum of Ball A is directed along the incline.
- Since momentum is a vector quantity, the direction matters.

6. **Conclusion**:
- Although the magnitudes of the momentum of both balls are equal, their directions are different.
- Therefore, the momenta of the two balls are not equal.

### Final Statements:
- **Statement 1** is true: The momentum of the two balls is not equal.
- **Statement 2** is also true: Momentum depends on both the magnitude and direction of velocity.