A body is falling freely under the action of gravity alone in vacuum. Which of the following quantities remain constant during the fall ?

Options

A. Kinetic energy
B. Potential energy
C. Total mechanical energy
D. Total linear momentum

Explanation

To determine which quantities remain constant during the free fall of a body under the action of gravity alone in a vacuum, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding Free Fall**:
- A body falling freely means it is only under the influence of gravity, with no air resistance or other forces acting on it. The acceleration due to gravity (g) is constant.

2. **Potential Energy (PE)**:
- The potential energy of the body is given by the formula \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.
- As the body falls, its height \( h \) decreases, which means the potential energy decreases continuously.

3. **Kinetic Energy (KE)**:
- The kinetic energy of the body is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( v \) is the velocity of the body.
- As the body falls, its velocity increases due to the constant acceleration of gravity. Therefore, the kinetic energy increases continuously.

4. **Total Mechanical Energy (TME)**:
- The total mechanical energy is the sum of potential energy and kinetic energy: \( TME = PE + KE \).
- Although potential energy decreases and kinetic energy increases, the total mechanical energy remains constant. This is because the decrease in potential energy is exactly balanced by the increase in kinetic energy.

5. **Total Linear Momentum**:
- Linear momentum (p) is given by \( p = mv \).
- As the body falls, its velocity \( v \) increases, which means the linear momentum also increases. Therefore, linear momentum does not remain constant.

### Conclusion:
- The quantities that remain constant during the fall are:
- **Total Mechanical Energy** (TME)

### Summary of Results:
- **Potential Energy**: Decreases
- **Kinetic Energy**: Increases
- **Total Mechanical Energy**: Constant
- **Total Linear Momentum**: Increases

### Final Answer:
The quantity that remains constant during the fall is **Total Mechanical Energy**.

---