The Law of Change 
Because of the extreme restrictions placed on the systems and interactions described by the One-Dimensional Motion with Constant Velocity model, the Law of Change for the model is rather simple. The mathematical definition of velocity (for one-dimensional motion) is:
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\begin{large}\[ v \equiv \frac{dx}{dt}\]\end{large} |
If v is a constant, this equation can be straightforwardly integrated:
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\begin{large}\[ \int_{t_{A}}^{t_{B}} v\:dt = \int_{x_{A}}^{x_{B}} dx \]\end{large} |
which (after algebraic rearrangement) gives:
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\begin{large}\[ x_{B} = x_{A} + v(t_{B} - t_{A})\]\end{large} |
where:
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\begin{large}\[ x_{A} \equiv x(t_{A}) \]\[x_{B} \equiv x(t_{B})\]\end{large} |
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It is rare for physics problems to specify an initial time for a motion, but rather they will usually specify an elapsed time. For instance, instead of saying "a car began a trip at 10:05 AM and drove until 10:15 AM", the problem will usually specify only that the car drove "for 10 minutes". Elapsed time is equivalent to the difference tB - tA. |