Question

Define K.E. Give its example and derive $KE=\frac{1}{2}m{v}^2$.

Answer 1

Definition of kinetic energy

The energy possessed by a body due to its motion is termed it is kinetic energy.

Kinetic Energy is given as, $KE=\frac{1}{2}m{v}^2$ [where $m$ is mass and $v$ is velocity]

Example of kinetic energy

Any object which possesses velocity possesses kinetic energy. For example:

1. A ball falling from a height.
2. A moving car
3. A bullet fired from a gun

Derivation

Consider an object of mass $m$ lying at rest on a table. Suppose a force $F$ acts on the object and the object moves through a distance $S$ under the effect of the applied force.

Now, work done is given as,

$W=FS$

where, $F$ is force and $S$ is distance.

When the force is applied, the body acquires a velocity. Let $a$ be the acceleration of the body.

Using the third equation of motion,

$v^2=u^2+2aS$

where $v$ is final velocity, $u$ is initial velocity, $a$ is acceleration and $S$ is the distance traveled by the object.

Using the above equation, an equation for distance can be obtained as,

$S=\frac{v^2-u^2}{2a}$

Since the body was at rest initially, the initial velocity of the body will be zero, that is, $u=0$. Therefore,

$$\begin{gathered} S=\frac{v^2-0^2}{2a} \\ S=\frac{v^2}{2a} \end{gathered}$$

Now, we know that force is given as,

$F=ma$

where $m$ is mass and $a$ is acceleration

Substituting the equation of force and distance in the equation for work done, we get.

$$\begin{gathered} W=FS \\ W=\left(ma\right)\left(\frac{v^2}{2a}\right) \\ W=\frac{1}{2}m{v}^2 \end{gathered}$$

According to the work-energy theorem, work done by a body equals the change in its kinetic energy.

Since, $KE=W$, therefore, $KE=\frac{1}{2}m{v}^2$