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

Correct Option: Total mechanical energy

Let us analyze the behavior of a freely falling body in a vacuum step-by-step to understand why the total mechanical energy remains constant.

1. Understanding Free Fall in a Vacuum
When a body falls freely in a vacuum, the only force acting on it is the force of gravity. Because there is a vacuum, air resistance and friction are completely absent. The force of gravity is a conservative force.

2. Analysis of the Quantities

• Kinetic Energy (K):
As the body falls, it accelerates downwards under gravity, causing its velocity (v) to increase continuously. The kinetic energy is given by:
$K=\frac{1}{2}m{v}^2$
Since velocity increases, the kinetic energy also increases during the fall, meaning it is not constant.

• Potential Energy (U):
As the body falls, its height (h) above the reference ground level decreases. The gravitational potential energy is given by:
$U=mgh$
Since height decreases, the potential energy decreases during the fall, meaning it is not constant.

• Total Linear Momentum (p):
Linear momentum is the product of mass and velocity:
$p=mv$
Since velocity increases during the fall, the linear momentum also increases, meaning it is not constant.

• Total Mechanical Energy (E):
According to the law of conservation of mechanical energy, when only conservative forces (like gravity) do work on a system, the total mechanical energy (the sum of kinetic energy and potential energy) remains conserved (constant) at every point of the motion:
$E=K+U=\text{Constant}$
As the body falls, potential energy is continuously transformed into kinetic energy at an equal rate, keeping their sum constant throughout the entire motion.

Therefore, the quantity that remains constant during the fall is the Total mechanical energy.