Equation of Motion and its Application

Introduction to Equations Of Motion

In this article, we will learn how we can relate quantities like velocity, time, acceleration and displacement provided the acceleration remains constant. These relations are collectively known as the equation of motion. There are three equations of motion. There are three ways to derive the equation of motion and here we are going to derive with the help of graph.

First Equation of Motion

First equation of motion relates velocity, time and acceleration. Now in ∆uxy,

tanθ = \( \frac {xy}{uy}\)

tanθ = \( \frac {v~-~u}{t}\)

We also know that tanθ is nothing but the slope and slope of v – t graph represents acceleration.

⇒ v = u + at ———– (1)

This is the first equation of motion where,

v = final velocity

u = initial velocity

a = acceleration

t = time taken

Second Equation of Motion

Now coming to the second equation of motion, it relates displacement, velocity, acceleration and time. The area under v – t graph represents the displacement of the body.

In this case,

Displacement = Area of trapezium (ouxt)

S = \( \frac 12 \)  x sum of parallel sides x height

S = \( \frac 12 \)  x (v + u) x t ———- (2)

We can substitute v in terms of others and get final equation as:

S = ut + \( \frac 12~ at^2\)

Where symbols have their usual meaning.

Third Equation of Motion

The third equation of motion relates velocity, displacement and acceleration. Using same equation (2),

S = \( \frac 12 \)  x (v + u) x t

Using equation (1) if we replace t we get,

S = \( \frac 12 \)  x \( (v + u) \) x \( \frac{(v-u)}{a}\)

S = \( \frac {(v^2~-~ u^2)}{2a}\)

\( v^2\)  = \(  u^2~ +~ 2as\)

The above equation represents our third equation of motion.

Applications of Equation Of Motion

So now that we have seen all the three equations of motion we can use them to solve kinematic problems. We just have to identify what all parameters are given and then choose the appropriate equation and solve for the required parameter.

The equations of motion is also used in the calculation of optical properties.