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Linear motion is when an object moves on a straight line.There are two types of linear motion: uniform linear motion (when the acceleration is zero and the velocity is constant) and non uniform linear motion (when the acceleration isn't zero and the velocity is varriable).

Uniform Linear Motion

In uniform linear motion the velocity is constant so the kinematic equations are:

$ v = \frac{\Delta x}{\Delta t} $ (1.1)

$ x = v\Delta t $ (1.2)

$ t = \frac{\Delta x}{v} $ (1.3)

Non-Uniform Linear Motion

In non-uniform motion the velocity isn't constant since there is some acceleration $ a $.

The kinematic equations for constant acceleration are:

$ v_f = v_i + at $ (2.1)

$ x = v_it + \frac{1}{2}at^2 $ (2.2)

Here,

$ v_i $ is the initial speed,

$ v_f $ is the final speed,

$ t $ is the time

$ a $ is the acceleration (it can be either positive or negative)

Mathematical proof for the above formulas

$ a = \frac{dv}{dt} $

$ dv = adt $

$ \textstyle \int\limits_{v_i}^{v_f} dv = \textstyle \int\limits_{0}^{t} adt $

$ v_f - v_i = at $

$ v_f = v_i + at $ (2.1)


$ v = \frac{dx}{dt} $

$ \frac{dx}{dt} = v_i + at $ (from (2.1))

$ dx = (v_i + at)dt $

$ \textstyle \int\limits_{0}^{x} dx = \textstyle \int\limits_{0}^{t}(v_i + at)dt $

$ x = v_it + \frac{1}{2}at^2 $ (2.2)