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# Speed

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:

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