# Question #b63d2

Mar 14, 2017

Energy possessed by an object due to its motion is termed as its Kinetic Energy.

It follows that if there is no motion, or the object is stationary, its kinetic energy is zero.

From Law of conservation of Energy we also know that energy can not be created. It however, can be changed from one form to another.

Derivation.
It is assumed mass is constant.
We know that work done in accelerating an object of mass $m$ through an infinitesimal time interval $\mathrm{dt}$ is given by the dot product of force $\vec{F}$ and infinitesimal displacement $\mathrm{dv} e c r$. Therefore total work done is
$W = \int \vec{F} \cdot \mathrm{dv} e c r$
Using Newton's Second law of Motion
$\vec{F} = m \vec{a}$
the integral becomes

$W = \int m \vec{a} \cdot \mathrm{dv} e c r$
Writing acceleration in terms of velocity $\vec{v}$ we get
$W = m \int \frac{\mathrm{dv} e c v}{\mathrm{dt}} \cdot \mathrm{dv} e c r$
Rearranging we get
$W = m \int \frac{\mathrm{dv} e c r}{\mathrm{dt}} \cdot \mathrm{dv} e c v$
We know that $\frac{\mathrm{dv} e c r}{\mathrm{dt}} \equiv \vec{v}$, therefore our integral becomes
$W = m \int \vec{v} \cdot \mathrm{dv} e c v$

Assuming that the object was at rest at time $t = 0$, we integrate from $t = 0$ to $t = t$ as the work done by the force to bring the object from rest to velocity $v$ is equal to the work necessary to reverse the process.
$W = m \int \vec{v} \cdot \mathrm{dv} e c v$
$W = m \left[\frac{1}{2} {v}^{2} + C\right]$
where $C$ is constant of integration.

As work done is zero when $v = 0$, $\implies C = 0$
Hence, $W = K E = \frac{1}{2} m {v}^{2}$