Question

The potential energy of a freely falling object decreases progressively. Does this violate the law of conservation of energy. Why? Prove it

Answer 1

Step. 1 Formula used,

$\Rightarrow PE=m.g.h$

$\Rightarrow v^2-u^2=2gx$

$\Rightarrow KE={\displaystyle \frac{1}{2}}m{v}^2$

$m$ is the mass of the body, $h$ is the height

$PE$ is the potential energy

$KE$ is the kinetic energy

$v$is the final velocity

$u$is the initial velocity

$g$is the gravity

We consider the body dropped at a distance $x$

Step. 2 Calculate potential energy,

Consider a body of mass $m$, at a height $h$; its potential energy is given by,

$\Rightarrow PE=m.g.h$

Step. 3 Calculate new potential energy,

Let's consider the body dropped at a distance $x$ in time $t$, then its new potential energy will be,
$\Rightarrow P{E}_t=mg(h-x)$

The loss in potential energy is
$\Rightarrow PE-P{E}_t=mgh-mg(h-x)=mgx$………..Equation $1$

Step. 4 Calculate the gain in kinetic energy of the body,

The body attained velocity in time t can be written in terms of Newton’s third law of motion as
$\Rightarrow v^2-u^2=2gx$
Initial velocity $\left(u\right)$ =$0$,
$\Rightarrow v^2=2gx$……….Equation $2$
Now, since the body has gained kinetic energy with gain in velocity, hence the gain in kinetic energy of the body is,
$\Rightarrow KE={\displaystyle \frac{1}{2}}m{v}^2$

From eq $2$,
$\Rightarrow KE={\displaystyle \frac{1}{2}}m\times 2gx=mgx$
Substituting the value of $mgx$ from eq $1$, we get
$\Rightarrow KE=PE-P{E}_t$
$\Rightarrow PE=KE+P{E}_t$
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 process does not violate the law of conservation of energy. This is because when the body falls from a height, then its potential energy changes into kinetic energy progressively.

A decrease in the potential energy is equal to an increase in the kinetic energy of the body.

During the process, the total mechanical energy of the body remains conserved. Therefore, the law of conservation of energy is not violated.