Question

Derive an expression for the kinetic energy of an object of mass $m$ moving with a velocity $v$.

Answer 1

Step 1: Data provided,

$$\begin{gathered} mass=m \\ velocity=v \end{gathered}$$

Step 2: Formula used and Calculation:

$$\begin{gathered} v^2-u^2=2aS................................3rdequationofmotion \\ W=F\times S.........................................Formulaofworkdone \\ F=ma............................................Formulaofforce \end{gathered}$$

Let an object of mass $m$ be at rest. Thus, initial velocity, $u=0$. It is lying on a smooth surface. A constant force $F$ acts on the object such that the acceleration produced in the body is $a$. Let the body move with a velocity after a displacement ‘S’.

Then,

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

Now, the work done by the force in displacing the object is given by

$$\begin{gathered} W=F\times S \\ =ma\times S \\ =m\times \frac{v^2}{2S}\times S \\ =\frac{1}{2}m{v}^2 \end{gathered}$$

This work done is stored as kinetic energy. So the kinetic energy of the object is given by

$K=\frac{1}{2}m{v}^2$