# Why do we need instantaneous speed?

Because instantaneous speed affects physics. Imagine a wall $10~\textrm m$ in front of you. You walk towards it smoothly over a timeframe of, say, $20~\textrm s$, and without getting slower, you walk into the wall. You'll feel a slight bonk, but nothing serious is going to happen. Now imagine the same 20 seconds going differently: You wait for 17 seconds, then you sprint towards the wall at full speed.

Both scenarios will give you the same average speed over the 20 seconds, but you better be wearing a helmet for the second one. The difference lies in the fact that the instantaneous speed at the end of the 20 second interval is different. It's a quantity that affects things. So it makes sense to talk about it.

It is really simple:

Average speed is as good as instantaneous speed **only if speed does not change with time**. If you are studying a body with rapidly changing speed then using average speed to describe that body gives you an inaccurate depiction of the phenomena taking place.

Of course we can only truly measure average speeds, but measuring those over really short periods of time helps getting us closer to the true value of the instantaneous speed. In fact remember that the *definition* of instantaneous speed ($v_i$) is precisely average speed ($v_a(\Delta t)$) measured over an interval of time that *tends* to zero:

$$\lim _{\Delta t \to 0} v_a(\Delta t)=v_i$$

Instantaneous speed is what the police officer fines you for. On a trip to the bakery it doesn't really matter that your average speed is 40 km/hr if you during the trip reached 120 km/hr for just a moment.

In some scenarios an average is useful. In others the instantaneous speed is useful. An average is lacking full detail but "represents" the full trip concisely,** whereas having the instantaneous speed at every moment is accurate but a huge amount of data. Which you will be using depends on what you need and thus varies form case to case.

** ^{Also note that there are various types of averages with different ways of "representing" an amount of data in one single and more tangible number, such as arithmetic average or mean, geometric average, RMS value and the like.}