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Explained where instantaneous velocity comes from
'''Instantaneous velocity''' is the [[velocity]] of an object at a single instant in time.
The '''instantaneous velocity''' of an object is the [[velocity]] at a particular instant in time and is a [[vector]] quantity. It can be defined by considering the [[average velocity]] of a body. If a body moves a distance <math>\Delta \vec{r}</math> in a time <math>\Delta t</math>, then its average velocity over <math>\Delta t</math> is:
<math>
v_{average} = \frac{\Delta \vec{r}}{\Delta t}
</math>
As velocity is [[displacement]] divided by time. As the time interval <math>\Delta t</math> becomes smaller and smaller, this average velocity approximates better and better the velocity of the object at which the time interval is centred. To get the exact velocity at some time t, the limit of <math>\Delta t</math> going to 0 is taken:
<math>
v(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d \vec{r}}{dt}
</math>
In this limit, the instantaneous velocity becomes the [[derivative]] of the displacement. In a similar way, the instantaneous speed of an object is the derivative of the object's [[distance]] and as such is the gradient of a distance-time graph.
[[Category:Mechanics]]
[[Category:Mechanics]]