Question

What is kinetic energy? Derive an equation for the kinetic energy of a body of mass 'm' moving at a speed 'v'.

Answer 1

Recall the concepts of Force, third Equation of motion, and the definition of work done and use the formulas of all these three to derive the equation of Kinetic Energy.

Complete step-by-step answer:
Kinetic energy of a body or an object is the energy that it possesses by virtue of its motion. It may also be defined as the work needed to accelerate a body of a given mass from rest to its desired velocity.

Derivation:
Let us consider an object of m which is at rest lying at rest. Clearly Since it is not doing any motion, hence net work done by it is zero and in turn its kinetic energy is also 0. ( by the definition)
Now let’s consider an object which was initially at rest but now travelling with a certain velocity and acceleration ( Let’s say a force F act’ s on it and displaces the object'S distance apart)
Net work done = Force × Displacement.
W= F × S
Let the work done on the object cause a change in its velocity from U to V and let ‘a’ be the acceleration. From Third equation of motion:
(Since the object was initially at rest)
Now,
W= F $\times$ S
$\Rightarrow$ W = m $\times$ a $\times$ S (F = ma)
$\Rightarrow$ W = $\dfrac{1}{2}m{v^2}$

We know that the kinetic energy of a body moving with a certain velocity is equal to work done on the object to acquire that velocity from rest. (From definition)
Therefore K = W = $\dfrac{1}{2}m{v^2}$
Hence derived.

Note: Kinetic energy is not always equal to total work done. Moreover, Total Work done is a sum of kinetic energy + potential energy.