Class 9 > Unit # 02: Kinematics > Equations of Motion

Talha's Physics Academy

Unit No. 2 Kinematics - Derivations of Equations of Motion

Introduction

Suppose an object moves with initial velocity $v_i$ in a time $t$ and covers a distance $S$ with a uniform acceleration $a$, and its final velocity becomes $v_f$.

(i) First Equation of Motion

According to the definition of acceleration, "The rate of change of velocity is called acceleration":

$a = \frac{v_f - v_i}{t}$

By rearranging the terms:

$at = v_f - v_i$

$v_f = v_i + at$

This equation is known as the First Equation of Motion.

(ii) Second Equation of Motion

From the definition of acceleration:

$v_f = v_i + at \quad \text{--- (1)}$

The average velocity ($v_{av}$) can be calculated as:

$v_{av} = \frac{v_i + v_f}{2} \quad \text{--- (2)}$

The total distance covered ($S$) is given by distance equals average velocity multiplied by time:

$S = v_{av} \times t$

Putting the value of $v_{av}$ from equation (2):

$S = \left(\frac{v_i + v_f}{2}\right) t$

Putting the value of $v_f$ from equation (1):

$S = \left(\frac{v_i + (v_i + at)}{2}\right) t$

$S = \left(\frac{2v_i + at}{2}\right) t$

$S = \left(\frac{2v_i t + at^2}{2}\right)$

$S = v_i t + \frac{1}{2}at^2$

This equation is known as the Second Equation of Motion.

(iii) Third Equation of Motion

From the definition of acceleration:

$v_f = v_i + at \quad \text{--- (1)}$

The average velocity is:

$v_{av} = \frac{v_i + v_f}{2} \quad \text{--- (2)}$

The distance covered by the body is:

$S = v_{av} \times t$

Putting the value of $v_{av}$ from equation (2):

$S = \left(\frac{v_i + v_f}{2}\right) t$

From equation (1), time $t$ can be expressed as:

$t = \frac{v_f - v_i}{a}$

Substituting the value of $t$ into the distance equation:

$S = \left(\frac{v_i + v_f}{2}\right) \left(\frac{v_f - v_i}{a}\right)$

$S = \frac{(v_f + v_i)(v_f - v_i)}{2a}$

$2aS = v_f^2 - v_i^2$

$2aS + v_i^2 = v_f^2 \quad \implies \quad v_f^2 - v_i^2 = 2aS$

This equation is known as the Third Equation of Motion.

© 2026 Talha's Physics Academy. All rights reserved.