The time rate of change of position.
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Mathematical Definition
\begin
[ \vec
= \frac{d\vec{r}}
]\end
One-Dimensional Velocity
Utility of the One-Dimensional Case
As with all vector equations, the equations of kinematics are usually approached by separation into components. In this fashion, the equations become three simultaneous one-dimensional equations. Thus, the consideration of motion in one dimension with acceleration can be generalized to the three-dimensional case.
Diagrammatical Representation
The one-dimensional version of the definition of velocity is:
\begin
[ v = \frac
]\end
The form of this definition implies that an object's velocity will equal the slope of the object's position versus time graph. Some example problems that make use of this fact are:
Example – Graphing Constant Velocity
To see the utility of the graphical representation of velocity, consider the following plots:
Consider the left plot. The abscissa is time, and the ordinate is position. This graph is giving the position of some object (called object A) as a function of time. Looking at the graph, we can see that for each second of time that elapses, the object changes its position by 2 meters. This is the same as saying that the slope of the left plot is 2 m / (1 s) or, more simply, 2 m/s. Object A, then, is moving with a speed of 2 m/s.
Contrast that with the middle plot. Object B is only changing its position by 1 meter every second. Thus, it is moving with a speed of 1 m/s.
Finally, look at the plot on the right. Object C is changing its position by 2 meters every second, and so it has a speed of 2 m/s. Note that objects A and C have the same speed. The graphs are different, however, because object C is moving in the negative direction.
That doesn't necessarily mean objects A and C are moving in different actual directions. The graphs may be made with respect to different coordinate systems (does the positive x direction point east? west? north?...).