## FANDOM

9 Pages

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)