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.

## 0 Comments