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Instantaneous speed

Speed at some instant, or assumed constant during a very short period of time, is called instantaneous speed. By looking at a speedometer, one can read the instantaneous speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m.

In mathematical terms, the instantaneous speed v {\displaystyle v} is defined as the magnitude of the instantaneous velocity v {\displaystyle {\boldsymbol {v}}}, that is, the derivative of the position r {\displaystyle {\boldsymbol {r}}} with respect to time: v = | v | = | r ˙ | = | d r d t |. {\displaystyle v=\left|{\boldsymbol {v}}\right|=\left|{\dot {\boldsymbol {r}}}\right|=\left|{\frac {d{\boldsymbol {r}}}{dt}}\right|\,.}

If s {\displaystyle s} is the length of the path (also known as the distance) travelled until time t {\displaystyle t}, the speed equals the time derivative of s {\displaystyle s}: v = d s d t. {\displaystyle v={\frac {ds}{dt}}.}

In the special case where the velocity is constant (that is, constant speed in a straight line), this can be simplified to v = s / t {\displaystyle v=s/t}. The average speed over a finite time interval is the total distance travelled divided by the time duration.