These equations are used to describe motion in a straight line with uniform acceleration. You must to be able to:

- select the correct formula
- identify the symbols and units used
- carry out calculations to solve problems of real life motion

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## Proof of 1^{st} equation

Consider a body initial moving with velocity “u”. After certain interval of time “t”, its velocity becomes “v”. Now Change in velocity = v – u

And by the definition of acceleration, rate of change of velocity is called acceleration.

Acceleration = rate of change of velocity

**a=v-u/t**

*a = acceleration in metres per second per second (m s*^{–2})*v = final velocity in metres per second (m s*^{–1})*u = initial velocity in metres per second (m s*^{–1})*t = time in seconds (s)*

**v = u + at** **Equation of motion 1**

## Proof of 2^{nd} equation

Consider a car moving on a straight road with an initial velocity equal to ‘u’. After an interval of time‘t’ its velocity becomes ‘v’. Now first we will determine the average velocity of body.

Average velocity = (Initial velocity + final velocity)/2

OR

V_{av} = (u + v)/2

but v = u + at

Putting the value of v

Vav = (u + u + at)/2

Vav = (2u + at)/2

Vav = 2u/2 + at/2

Vav = u + at/2

Vav = u + 1/2at…

we know that

S = Vav x t

Putting the value of ‘Vav’

S = [u + 1/2at] t

We get

s = ut + ½at^{2}

S = ut + 1/2at^{2}

*s = displacement in metres (m)**u = initial velocity in metres per second (m s*^{–1})*t = time in seconds (s)**a = acceleration in metres per second per second (m s*^{–2})

**s = ut + ½at ^{2}**

**Equation of motion 2**

## Proof of 3rd equation

Equation 1 *v* = *u* + *at*

squaring each side of eq (1) to give

*v*^{2} = (*u* + *at*)^{2}

*v*^{2} = *u*^{2} + 2*uat* + *a*^{2}*t*^{2}

*v*^{2} = *u*^{2} + 2*a*(*ut* + ½*at*^{2})

As *ut* + ½*at*^{2} = S

substitute in Equation 2 *v*^{2} = *u*^{2} + 2*as *

*v*^{2} = *u*^{2} + 2*as***Equation of motion 3**