# Mechanics: Chapter 1 – Motion in One Dimension

Motion in one dimension is fairly simple. I’ll break it down to you as briefly as possible. In one dimension, we essentially have one co-ordinate of position. Let’s call that x. Now velocity is the rate of change of x. Acceleration is the rate of change of the the velocity of x.

• Average velocity is $v = \frac{\Delta x}{\Delta t}$
• Instantaneous velocity is the derivative of x with respect to time i.e $v = \frac{dx}{dt} = \dot{x}$
• Average acceleration is $a = \frac{\Delta v}{\Delta t}$
• Instantaneous acceleration is the derivative of v with respect to time (in other words the second derivative of the position with respect to time) $a = \frac{dv}{dt} = \dot{v} = \ddot{x}$

In one dimension, the equations of motion assuming constant acceleration are as follows:

• $v = v_{0} + at$
• $x = x_{0} + ut + \frac{1}{2}at^2$
• $x = x_{0} + \frac{1}{2}(v_{0}+v)t$
• $v^2 = v_{0}^2 + 2as$