# Kinematics Equations

Kinematics is the branch of Physics which deals with the motion of objects. The motion of the objects can be described numerically, pictorially, graphically and also with the help of equations. Such set of equations are called Kinematics equations. They are usually set of very simple equations.

## The four types of kinematics equations.

Kinematics equations are a set of four equations which can be used to the predict an unknown physical parameter in the motion of an object provided certain other physical parameters are known. Kinematics equations gives us the relation between four physical quantities which are displacement, velocity, acceleration and time. If any of these three quantities are known then using the kinematics equations we can find the fourth unknown quantity. For example if we know the initial velocity and acceleration of a moving car, then we can predict the distance travelled by the car after a particular period of time.

## Derivation of the kinematics equations

Equation1 : Velocity - time relation vf = vi +at

Consider a particle moving with an initial velocity' vi'. After a time 't' the velocity of the particle changes to 'vf'. This change in velocity produces an acceleration 'a' which is given by

acceleration a = change in velocity / time = (vf - vi)/t

at = vf - vi

vf = vi + at

Equation 2 : Position - average velocity relation S = (vi + vf)t / 2

If the velocity of a particle is constant, the displacement = velocity x time

If it moves with uniform acceleration, the displacement is given by

S = average acceleration x time

since acceleration is uniform, average velocity = (vi + vf) / 2

Hence displacement S = (vi + vf)t / 2

Equation 3 : Position - time relation S = vi*t + 1/2at2

Consider the position velocity relation

if we put vf = vi + at in that we have

S = (vi + vi+at)*t / 2 = (2vi*t + at2)/2

or S = vi*t + 1/2at2

Equation 4 : Position - velocity relation vf2 - vi2 = 2aS

Acceleration = change in velocity / time

a = (vf - vi)/t

or vf - vi = at

Displacement S = average velocity x time = (vi + vf)*t / 2

or vf + vi =2S/t

so (vf - vi) * (vf + vi) = 2aS

or vf2 - vi2 = 2aS

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