The Five Equations

The set of two equations on the previous page are enough to solve any one-dimensional constant-acceleration kinematics problem. However, we can take those two equations and, by recombining them in different ways, end up with this set of five equations:

\begin{aligned} v_{f} & = v_{i} + a Δ t & no Δ x \\ Δ x & = \frac{1}{2} (v_{i} + v_{f}) Δ t & no a \\ Δ x & = v_{i} Δ t + \frac{1}{2} a Δ t^{2} & no v_{f} \\ Δ x & = v_{f} Δ t - \frac{1}{2} a Δ t^{2} & no v_{i} \\ v_{f}^{2} & = v_{i}^{2} + 2 a Δ x & no Δ t \end{aligned}

A warning: these are not five independent equations; we can still only solve for two of the five variables.

So why complicate things? Notice that each of the equations has exactly one of the five variables missing. To solve a kinematics problem for one of the variables, you need to be given three: that leaves one variable that you don't know and don't care about. If you choose the equation that does not include that "don't-know-don't-care" (DKDC) variable, then you can solve that equation directly for the variable you want to know.

Example

For example, if I know a car accelerates from 10m/s to 15m/s over a distance of 500m, and I want to know its acceleration, then my DKDC variable is time. The last equation above doesn't include time, and so if I use it I can solve directly for acceleration:

15^{2} = 10^{2} + 2 a (500) ⟹ a = 0.125 m / s^{2}