Question

The potential energy of a freely falling object decreases progressively. Does this violate the law of conservation of energy?
(A) Yes
(B) No
(C) Yes, at certain instants
(D) None of the above

Answer 1

Hint
By virtue of falling, the body loses its height but gains velocity, hence, the potential energy of the falling body gets converted into kinetic energy. Use the equations of potential energy $PE = mgh$ and kinetic energy $KE = \dfrac{1}{2}m{v^2}$ to compare the loss in potential energy and gain in Kinetic Energy.

Complete step by step answer
Consider a body of mass m, at a height h; its potential energy is given by,
$\Rightarrow PE = m.g.h$
Now, consider the body is dropped, and falls a distance x in time t, then its new potential energy will be,
$\Rightarrow P{E_t} = mg(h - x)$
Hence the loss in potential energy is
$\Rightarrow PE - P{E_t} = mgh - mg(h - x) = mgx$ ..... Equation 1
Now, we know that the velocity attained by the body in time t is given by Newton’s third law of motion as
$\Rightarrow {v^2} - {u^2} = 2gx$
Also, since the initial velocity, u of the body was zero,
$\Rightarrow {v^2} = 2gx$ ..................................................... Equation 2
Now, since the body has gained velocity, it has also gained Kinetic Energy, the gain in kinetic energy of the body is given by,
$\Rightarrow KE = \dfrac{1}{2}m{v^2}$
Substituting the value of the square of velocity from equation 2 we get,
$\Rightarrow KE = \dfrac{1}{{{2}}}m \times {2}gx = mgx$
Substituting the value of mgx from Equation 1 we get
$\Rightarrow KE = PE - P{E_t}$
By rearranging the terms we can write,
$\Rightarrow PE = KE + P{E_t}$
Therefore the initial potential energy of a body is the same as the new potential energy and gain in kinetic energy. Hence we can say, NO, the progressive decrease in the potential energy of a freely falling object doesn’t violate the law of conservation of energy.
Option (B) is correct.

Note
The total mechanical energy of the object in a free fall is constant or conserved, the potential energy gets converted into kinetic energy progressively. This does not take into account the air resistance. If we take air resistance into account, the mechanical energy of the object is converted into kinetic energy of the air molecules and heat, and even in that case, the energy of the system is conserved. Hence, there are no exceptions to the law of conservation of energy in Newtonian mechanics.