State and prove the law of conservation of energy in case of a freely falling body

Consider a body m lying at a position A at a height h, above the ground. As the body falls, its kinetic energy increases at the expense of its potential energy.

At point A

The body is at rest

The kinetic energy of a body at A = K_A

The potential energy of body at A = U_A

The total techanical energy of body at A = E_A

At point B

Suppose the body falls through a distance x and reaches B with velocity v. Then

u = 0

s = x

a = g

The kinetic energy of body at B = K_B

The potential energy of a body at B = U_B

The total energy of the body at B = E_B

At point C

Suppose the body reaches C (on the ground) with velocity v’. Then considering motion from A to C

u = 0

s = h

a = g

v = v’

Kinetic energy of body at C = K_C

Potential energy of body at C = U_C

Total mechanical energy of body at C = E_C

As a body falls freely under gravity P.E. decreases and K.E increases by equal amounts, however the, total mechanical energy remains constant.