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| title | Four Useful Equations | borderStyle
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| solid | | -inline{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 |
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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:
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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 |
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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"? |