When a body falls freely towards the Earth, then its total energy

Concept

For motion under a conservative force, mechanical energy is the sum E = K + U. When gravity is the only force doing work, changes in kinetic and gravitational potential energy are equal in magnitude and opposite in sign, so ΔE = ΔK + ΔU = 0.

Application

1. Let the body fall from height h1 to h2, where h1 > h2.

2. The change in gravitational potential energy is ΔU = mg(h2 − h1) = −mg(h1 − h2).

3. From the free-fall relation v22 − v12 = 2g(h1 − h2), the change in kinetic energy is ΔK = ½m(v22 − v12) = mg(h1 − h2).

4. Therefore, ΔK + ΔU = mg(h1 − h2) − mg(h1 − h2) = 0.

Cross-check

The work–energy theorem gives ΔK equal to the work done by gravity, while the change in potential energy is the negative of that work. Their sum is therefore unchanged.

Hence, during free fall (with air resistance neglected), the total mechanical energy remains constant.