## The Equations of Constant Acceleration Some of the graphical representations of motion had their origins with Galileo, but their algebraic form waited for the 17th and 18th centuries to become standard practice. We'll make light use of 4 formulas for constantly accelerated motion that relate: $x$ (or actually, $x-x_0 = \Delta x$), $t$ (which is really the interval, $t-t_0 = \Delta t$ but with the start time as $t_0=0$), $v_0$ (the initial velocity), $v$ (the final velocity), and $a$ (the acceleration). \begin{align} & \text{1. } \; x = \langle v \rangle \\ & \text{2. } \; v = v_0 + at \\ & \text{3. } \; x = x_0+v_0 t + \frac{1}{2} at^2 \\ & \text{4. }\; v^2 = v_0^2 + 2ax \end{align} We've not worked out the fourth equation in equation block 2.1 but it comes from an algebraic elimination of time between the first and third relations. The curves above originally came from these formulae for the particular acceleration, $a \to g=10$ m/s$^2$. ```{admonition}   Pens out! :class: warning Have a look at a deeper explanation by deriving all of these equations ``` While we owe Galileo for the acceleration due to gravity on earth, his conclusions work for any constantly accelerated object, not just falling. So we would use the general acceleration, $a$ and when we’re dealing with things falling near the Earth, we’d use the specific acceleration. Let’s work out what the acceleration would be for cars that you may or may not drive.