# Equation of motion with the help of graph method

When a particle is moving with uniform acceleration along a straight line, then the mathematical equation by which the relationships of the quantities like velocity, acceleration, displacement etc. of such particle are expressed is called the equation of motion of the object.

**The following equations of motion are :**

v=u+at

s=ut+at²/2

v²=u²+2as

## Deduction of equations of motion with the help of graphical method

**1. Velocity-Time Relation (v=u+at)**

With the help of the above graph,

Suppose an object is move with uniform acceleration a. Its initial velocity is u(t=0), after time t the velocity of the particle becomes v.

Slope of straight line AC

Acceleration=Perpendicular/Base

a=BC/AB

a=(v-u)/t

at=v-u

**v=u+at**

**2. Position-Time Relation (s=ut+at²/2)**

With the help of the above graph,

distance traveled by a moving object

s=area enclosed between velocity time graph and the time axis

s=∆ABC+Area of ABDO

s=1/2×(AB×BC)+(AB×BD)

s=1/2.t(v-u)+t×u

s=1/2.t(at)+ut

s=1/2.at²+ut

**s=ut+1/2.at²**

**3. Position-Velocity Relation (v²=u²+2as)**

With the help of the above graph,

Distance traveled by a moving object at any instant t

s=(sum of parallel sides × distance between them)/2

s=[(AO+CD)×OD]/2

s=[(u+v)×t]/2

From relation (1)

v=u+at

t=(v-u)/a

s=[(u+v)×(v-u)/a]/2

s=(v²-u²)/2a

**v²=u²+2as**

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