Third Equation of Motion by Graphical Method

Here in this article, we will derive the third equation of motion by graphical method.

Formula for the third equation of motion:

Third equation of motion is given by the relation
$v^2=u^2+2as$
Where,
$v=$ final velocity
$u=$ initial velocity
$a=$ acceleration
$s=$ distance travelled
Note: - This equation along with other kinematics equations of motion are valid for objects moving with uniform acceleration.

Derivation of 3rd equation of motion by graphical method:

To derive 3rd equation of motion we will make following assumptions
•     Object under consideration is moving with acceleration $a\,\, m/s^2$
•     At time $t=0$ object have some initial velocity. Let’s denote it by $u\,\, m/s$
•     At time $ts$ object have some final velocity. Let’s denote it by $v\,\, m/s$
•     Total distance covered by object in time $t$ seconds is $s$ meters.
Object is moving with a uniform acceleration “a” along a straight line. The initial and final velocities of the object at time $t = 0$ and $t = t\,\, s$ are $u$ and $v$ respectively. During time $t$, let $s$ be the total distance travelled by the object.
Figure given below show the velocity-time graph for the object whose initial velocity is $u$ at time $t=0$ and velocity $v$ at time $t$.

We know that the distance covered by the object moving with uniform acceleration is given by the area under the velocity-time graph. Here area under the velocity-time graph is equal to area of trapezium OPQS

∴ Area of trapezium OPQS $= \frac{1}{2}\text{(Sum of Parallel Slides + Distance between Parallel Slides)}$
Or, $s=\frac{OP+SQ}{2}\times PR$

We have knowledge about acceleration of the moving object. So,
Acceleration  $a=\frac{\text{Change in velocity}}{time}=\frac{QR}{PR}$
Or,
$PR=\frac{QR}{a}$
From graph we can see that
$QR=v-u$
$t=PR=\frac{v-u}{a}$
From above figure we can see that
$OP=u$
$SQ=v$
$OP+SQ=u+v$
Substituting these values, we get
$s=\left(\frac{u+v}{2}\right)\times\left(\frac{v-u}{a}\right)=\frac{v^2-u^2}{2a}$
Rearranging it we get
$v^2=u^2+2as$
Which is third equation of motion.