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:
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{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} |
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It is clear from these equations that there are seven possible unknowns in a given problem involving motion between two points with constant acceleration:
- initial time ( ti )
- final time ( tf )
- initial position ( xi )
- final position ( xf )
- initial velocity ( vi )
- final velocity ( vf )
- acceleration (constant) a
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Remember that you will usually have the freedom to define the initial time and the initial position by setting up a coordinate system. |
Looking at the four equations, you can see that each is specialized to deal with problems involving specific combinations of these unknowns.
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| title | An Exercise in Derivation |
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Because the initial position and initial time can generally be arbitrarily chosen, it is often useful to rewrite all these equations in terms of only five variables by defining:
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{center}{latex}\begin{large}\[ \Delta x \equiv x_{f}-x_{i} \]\[\Delta t \equiv t_{f}-t_{i}\]\end{large}{latex}{center} |
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"? |