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

h3.  One Simple Law 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.  If [velocity] is constant, it can be found mathematically by the expression:

{latex}\begin{large}\[ \int_{t_{A}}^{t_{B}} v\:dt = \fracint_{x - x_{iA}}^{t-tx_{iB}} dx \]\end{large}{latex}

where:

{latex}

\begin{large}\[ x \equiv x(t) \]\[x_{_{B} = x_{A} + v(t_{B} - t_{A})\]\end{large}

\begin{large}\[ x_{Ai} \equiv x(t_{iA}) \]\[v(t) = vx_{B} \equiv x(t_{iB}) = v\]\end{large}{latex}

{note}It is important to note that the Law of Change for this model is actually usually simpler than even 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, it is usual to _choose_ the origin of the time coordinate such that _t_~i~ = 0, which simplifies the equation.{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.