Find the derivation of $E = m{c^2}$ ?

Hint: To solve this question, we need to find two relation between energy, force and momentum. The two definitions that will help us to get these two equation are: work is the product of the force applied on a body and the displacement caused by it, and, momentum is the product of mass and velocity of a body.

Complete step by step answer:
So, first we need to find the two relations between energy, force and momentum.
First relation: we know that work done is the product of force and displacement. We also know that energy is the capacity to do work, so energy can also be defined as the product of work done and the displacement taken place.
$E = F \times d$

In this case the object moves at the speed very close to the speed of light in vacuum $c$. So, the body travels a distance equal to $c$ in unit time. So, the above equation can also be written as,
$E = F \times c.......(1)$
Second relation: We know that the momentum gained by a body can be defined as the product of force and the time during which this force acts.
$p = F \times t$

The unit time during which this particular force acts on the body, the mass increases by an amount $m$ and the velocity remains constant at very close to the speed of light $c$. We also know that momentum is defined as the product of mass and velocity of the body. So, the equation can be written as,
$F = m \times c.......(2)$
On putting the value of force from equation (2) in equation (1), we get,
$E = m \times c \times c$
$\therefore E = m{c^2}$
So, this is the required equation.

Note: The $E = m{c^2}$ states that matter and energy are both inter-convertible.Energy can be converted into huge objects of large masses and at the same time, and at the same time, hug masses of objects can be used to create a large amount of energy.