An Abundance of Laws of Change    

Integrating the definitions of acceleration and velocity for the special case that acceleration is constant leads to four expressions that are commonly encountered in descriptions of motion with constant acceleration:

{center}{latex}\begin{large}\[ v_{f} = v_{i} + a(t_{f}-t_{i})\]\[x_{f} = x_{i} + \frac{1}{2}(v_{f}+v_{i})(t_{f}-t_{i})\]\[x_{f}=x_{i}+v_{i}(t_{f}-t_{i})+\frac{1}{2}a(t_{f}-t_{i})^{2}\]\[v_{f}^{2}=v_{i}^{2}+2a(x_{f}-x_{i})\]\end{large}{latex}{center}

It is clear from these equations that there are seven possible unknowns in a given problem involving motion between two points with constant acceleration:

1. initial time (t_i)
2. final time (t_f)
3. initial position (x_i)
4. final position (x_f)
5. initial velocity (v_i)
6. final velocity (v_f)
7. acceleration (constant) a

Looking at the four equations, you can see that each is specialized to deal with problems involving specific combinations of these unknowns.