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 (ti)
  2. final time (tf)
  3. initial position (xi)
  4. final position (xf)
  5. initial velocity (vi)
  6. final velocity (vf)
  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.

Training Flight ()

Because the initial position and initial time can be arbitrarily chosen, it is possible to rewrite all these equations in terms of only 5 variables by defining:
{latex}\begin{large}\[ \Delta x \equiv x_{f}-x_{i} \]\[\Delta t \equiv t_{f}-t_{i}\]\end{large}{latex}
If you replace the initial and final positions and times with these "deltas", then each of the equations given above involves exactly four unknowns.  Interestingly, the four equations represent all but _one_ of the unique combinations of four variables chosen from five possible unknowns.  Which unique combination is missing?  Can you derive the appropriate "fifth equation"?