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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.  IfThe mathematical definition of [velocity] is constant, it can be found mathematically by the expression(for one-dimensional motion) is:

{latex}\begin{large}\[ v \equiv \frac{dx}{dt} = \frac{x_{f} - x_{i}}{t_{f}-t_{i}}\]\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}

{warning}Note that the first equality is the _definition_ of velocity, which _always_ holds.  The second equality is _*only*_ true if the velocity is _*constant*_.{warningwhich (after algebraic rearrangement) gives:

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

where:

{latex}\begin{large}\[ x_{fa} \equiv x(t_{fa}) \]\[x_{ib} \equiv x(t_{ib})\]\end{large}{latex}

{note}It is important to note that the Law of Change for this model is usually even simpler than the form written above!  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".  Thus,Elapsed ittime is usualequivalent to the _choosedifference_ the origin of the time coordinate such that _t_~i~ = 0, which simplifies the equation.{_}t{~}b{~}{_} - {_}t{~}a{~}{_}{note}