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h3.  The Law of Change    [!copyright and waiver^SectionEdit.png!|Motion with Constant Velocity (Laws of Change)]

Because of the extreme restrictions placed on the [systems|system] and [interactions|interaction] described by the [One-Dimensional Motion with Constant Velocity|1-D Motion (Constant Velocity)] [model], the [Law of Change] for the model is rather simple.  The mathematical definition of [velocity] (for one-dimensional motion) is:

{latex}\begin{large}\[ v \equiv \frac{dx}{dt}\]\end{large}{latex}

If _v_ is a constant, this equation can be straightforwardly integrated:

{latex}\begin{large}\[ \int_{t_{a}}^{t_{b}} v\:dt = \int_{x_{a}}^{x_{b}} dx \]\end{large}{latex}

which (after algebraic rearrangement) gives:

{latex}\begin{large}\[ x_{b} = x_{a} + v(t_{b} - t_{a})\]\end{large}{latex}

where:

{latex}\begin{large}\[ x_{a} \equiv x(t_{a}) \]\[x_{b} \equiv x(t_{b})\]\end{large}{latex}

{note}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_ {_}t{~}b{~}{_} - {_}t{~}a{~}{_}.{note}