In the motion of a projectile freely under gravity, its

Options

A. total mechanical energy is conserved
B. momentum is conserved
C. mechanical energy and momentum both are conserved
D. None is conserved

Explanation

To solve the question regarding the motion of a projectile freely under gravity, we will analyze each option provided and determine whether it is true or false.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- A projectile is an object that is thrown into the air with an initial velocity and is subject to the force of gravity. The only force acting on it during its flight is gravity, which acts downwards.

2. **Analyzing Option A: Total Mechanical Energy is Conserved**:
- In projectile motion, the total mechanical energy (kinetic energy + potential energy) remains constant if we neglect air resistance. As the projectile rises, its kinetic energy decreases while its potential energy increases, and vice versa as it falls. Therefore, total mechanical energy is conserved.
- **Conclusion**: Option A is **correct**.

3. **Analyzing Option B: Momentum is Conserved**:
- Momentum is conserved in a system only when there are no external forces acting on it. In the case of projectile motion, gravity acts as an external force throughout the motion. Therefore, the momentum of the projectile is not conserved.
- **Conclusion**: Option B is **incorrect**.

4. **Analyzing Option C: Mechanical Energy and Momentum Both are Conserved**:
- Since we have established that momentum is not conserved due to the presence of an external force (gravity), this option is also incorrect. While mechanical energy is conserved, momentum is not.
- **Conclusion**: Option C is **incorrect**.

5. **Analyzing Option D: None is Conserved**:
- Since we have already determined that total mechanical energy is conserved, this option is also incorrect.
- **Conclusion**: Option D is **incorrect**.

### Final Answer:
- The correct option is **A: Total mechanical energy is conserved**.