# Definition: Average Velocity

The average velocity is the ratio between the displacement of an object in space $$\mathbb R^n$$ (e.g. for $$n=3$$), given by the difference of the departure and destination point $$\Delta x:=x_2-x_1$$, and the duration of its movement, given by the difference of the departure and destination time $$\Delta t:=t_2-t_1$$: $\bar {v}=\frac{\text{destination point}-\text{departure point}}{\text{destination time}-\text{departure time}}=\frac {\Delta x}{\Delta t}.$

For a time interval $$I\subset \mathbb R$$, it is convenient to model the position of the object in space $$x$$ as a function of time $$t$$: $t:\cases{I\to\mathbb R^n\\t\to x(t),}$ called a curve (or the trajectory function) of the object in space. At the departure time $$t\in I$$, the object is located at the "departure" position $$x(t)$$. After the time period $$h > 0$$ with $$t+h\in I$$, the object will move to the position $$x(t+h)$$. In this case, the average velocity of the object is given by

$\bar {v}=\frac {x(t+h)-x(t)}{h}.$

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