Instantaneous velocity and instantaneous speed
Average velocity yields only an overall description of the rate of motion but no information on the details. Consider a biker who travels the distance between two neighbouring towns in a given time. We can calculate her average velocity – let us assume that this is positive. Even if we know this average velocity, we have no information whether she stopped or even turned back (that is, travelled with negative velocity) in between the starting point and the endpoint of her path.
This information is given by the quantity called the instantaneous velocity – that is, velocity at the moment. Whilst average velocity characterises the whole of the motion as a longer process, instantaneous velocity describes the motion only for the given instant, and it may keep changing from moment to moment.
How can we obtain the instantaneous velocity? We have to calculate the ratio of displacement to elapsed time for a very short interval. But how short? The answer is as short as possible without actually being zero. The exact mathematical operation is the limit as the length of the interval approaches zero, which turns the ratio of differences into a derivative.
Instantaneous velocity
The instantaneous velocity \(v_x\) equals the limit of the ratio of the displacement \(\Delta x\) to the time interval \(\Delta t\) during which that displacement occurs as \(\Delta t\) approaches zero:
\[v_x := \lim\limits_{\Delta t \to 0}\frac{\Delta x}{\Delta t}.\]
Notice that this is the first derivative of the position with respect to time:
\[v_x := \frac{\mathrm{d}x}{\mathrm{d}t}.\]
Instantaneous velocity may change from instant to instant, which means \(v_x\) is a function of time: \[v_x = v_x(t).\]
Instantaneous velocity is a vector quantity.
The SI unit of instantaneous velocity is the metre per second:
\[\left[v_x\right] = 1\,\frac{\textrm{m}}{\textrm{s}}.\]
Instantaneous speed
The instantaneous speed \(v_x^*\) equals the magnitude of the instantaneous velocity:
\[v_x^* = \left|v_x\right|.\]
(Remember, the magnitude of the average velocity is not equal to the average speed! For the instantaneous quantities, the time interval is so short that the difference between the magnitude of the displacement and the distance travelled vanishes, that is why the equality of magnitudes holds true.)
Instantaneous speed is a scalar quantity.
The SI unit of instantaneous speed is the metre per second:
\[\left[v_x^*\right] = 1\,\frac{\textrm{m}}{\textrm{s}}.\]