Question

Prove that the total mechanical energy of a freely falling body is conserved.

Answer 1

During the fall, the body is at a position B. The body has moved a distance x from A. Thus we have seen that sum of potential and kinetic energy of freely falling body at all points remains same. Under the force of gravity, the mechanical energy of a body remains constant.

Answer 2

It may be shown that in the absence of external frictional force the total mechanical energy of a body remains constant.Let a body of mass m falls from a point A, which is at a height h from the ground as shown in fig.
At A,
Kinetic energy kE = 0Potential energy Ep = mghTotal energy E = Ep + Ek = mgh + 0= mgh
During the fall, the body is at a position B. The body has moved a distance x from A.
At B,
velocity v2 = u2 + 2as
applying, v2 = 0 + 2ax = 2ax
Kinetic energy Ek = 1/2 mv2 = 1/2 m x 2gx = mgxPotential energy Ep = mg (h – x)Total energy E = Ep + Ek = mg (h-x) + mgx = mgh – mgx + mgx= mgh
If the body reaches the position C.
At C,
Potential energy Ep = 0Velocity of the body C isv2 = u2 + 2asu = 0, a = g, s = happlying v2 = 0 + 2gh = 2gh
kinetic energy Ek =1/2 mv2=1/2 m x 2gh= mgh
Total energy at C                      E = Ep + Ek                     E = 0 + mgh                     E = mghThus we have seen that sum of potential and kinetic energy of freely falling body at all points remains same. Under the force of gravity, the mechanical energy of a body remains constant.