The distance from school to bagel store is 2 kilometres, $\Delta s=s_\text{end}- s_\text{start}=2000\,\mathrm m$. Your coffee break is 500 seconds long (a bit below 10 minutes), $\Delta t= t_\text{end}- t_\text{start}=500\,\mathrm s$. How fast must you go on your bike to make it?

Let’s give the idea of ‘how fast’ a name: why not **speed** or **velocity **with the symbol $v$.^{1} When we say: ‘how fast’, we usually mean: ‘how far in how much time’. When someone asks ‘how fast’, you must in your case answer: ‘$2000\,\mathrm m$ in $500\,\mathrm s$’.

- Or you could answer: ‘$1000\,\mathrm m$ for every $250\,\mathrm s$’.
- Or ‘$100\,\mathrm m$ for every $25\,\mathrm s$’:
- Or ‘$4\,\mathrm m$ for every $1\,\mathrm s$’, in short ‘$4\,\mathrm m$ per $\mathrm s$’.

All these are different ways of telling the same speed. The simplest version is obviously to say: ‘$4\,\mathrm m$ per $\mathrm s$’. And we can get to that value by simply *dividing* the original values:

$$v_\text{average}=\frac{\Delta s}{\Delta t}= \frac{2000\,\mathrm m}{500\,\mathrm s}=4\,\mathrm{m/s}$$

The ‘$4\,\mathrm m$ per $\mathrm s$’ is here written in short as: $4\,\mathrm{m/s}$ where the $\mathrm{m/s}$ is a *combined* unit: ‘metres per second’.

But you stopped for red light along the way and went slower uphill and a bit faster downhill. Naturally, you didn’t drive $4\,\mathrm{m/s}$ the whole time, only on **average**. The speed has along the way been a bit higher and lower now and then, but the **average** speed is $4\,\mathrm{m/s}$.

Now, is an ‘average’ at all useful? Wouldn’t it be better with the *actual *speed at a specific moment in time, rather than the *average* speed over a duration of time? Well, sometimes yes, sometimes no:

- If you ask the driver: ‘how fast are we going?’, then you want the actual speed right now.
- But if you ask: ‘how fast did you go last time?’, then he can’t answer with just one actual speed, because he
*didn’t*drive with just one fixed speed all the time.

Here in the latter case, the average speed might be the better answer, because it “represents” the entire trip in a single value. An ‘average’ can thus be used as a “simplified” version, when details are not important.

References:

- ‘
**Sears and Zemansky’s Univesity Physics with Modern Physics**’ (book),*Hugh D. Young & Roger A. Freedman*, Pearson Education, 13th ed., 2012