# Equations of Motion | Definition, Formula

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## What is Equations of Motion

The equations of motion describe the motion of an object in terms of its displacement, velocity, and acceleration. These equations are fundamental in classical mechanics and are commonly used to analyse and predict the motion of objects, especially under the influence of forces. There are three basic equations of motion, which relate the initial velocity U and final velocity V, acceleration a, displacement s, and time t. The equations are as follows:

### Equations of Motion For Uniform Acceleration

Motion is the dynamic state characterized by the evolving position of an object over time. It finds expression through fundamental quantities such as displacement, distance, velocity, acceleration, time, and speed. Everyday activities like jogging, driving a car, or even a leisurely walk exemplify instances of motion. The interconnections among these quantities are encapsulated in what we refer to as the equations of motion. These equations serve as the mathematical framework to comprehend and describe the intricate relationships governing the motion of objects in our physical world.

In situations characterized by uniform acceleration, we rely on three fundamental equations known as the laws of constant acceleration. These equations play a pivotal role in determining key components such as displacement (s), initial (u) and final velocity (v) time (t), and acceleration (a). It is crucial to note that these equations are applicable exclusively when acceleration remains constant and the motion unfolds along a straight line. The trio of equations serves as a powerful tool for analysing and predicting the kinematics of objects subject to uniform acceleration in a linear trajectory.

#### The three equations are,

• v = u + at.
• v² = u² + 2as
• s = ut + ½at²
where,

s = displacement;
u = initial velocity;
v = final velocity;
a = acceleration;
t = time of motion.

These equations are referred as SUVAT equations where SUVAT stands for displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (T)

## First Equation Of Motion

v = u + at

Consider a body having initial velocity ‘u’. Suppose it is subjected to a uniform acceleration ‘a’ so that after time ‘t’ its final velocity becomes ‘v’. Now we now,

Acceleration = change in velocity/time

a= v-u/t

or v = u + at or v = at + u …..(i)

This equation is known as the first equation of motion.

## Second Equation Of Motion

s = ut + ½at²

Suppose a body has an initial velocity ‘u’ and uniform acceleration ‘a’ for time ‘t’ so that its final velocity becomes ‘v’. The distance traveled by moving body in time ‘t’ is ‘S’ then the average

velocity = (v + u)/2. Distance traveled = Average velocity × time

MATHEMATICAL DERIVATION OF EQUATIONS OF MOTION

S = ut + ½at²

This is the second equation of motion.

## Third Equation Of Motion

v2 - u2 = 2as

From the second equation of motion we have,

s = ut + 1/2at2…(i)

From first equation of motion, we have

v = u + at

⇒ at = v - u

⇒ t = v - u/a

Putting this value of ‘t’ in equation …(i)

We have

or v2 = u2 + 2as

or v2 - u2 = 2as

This is the third equation of motion

Where

v = final velocity

u = initial velocity

a = acceleration

s = distance traveled

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