Math and Life

# D e r t

D e r t

Distance equals rate times time.

That's how the 11 year old version of myself remembered the speed-distance equation: dert.

A handy little equation—it’s one of those things you actually use in real life. However, it has a way of fixing one particular way of thinking in our mind, and it does so very subtly.

Let's rearrange it for clarity:

See it yet? I certainly didn't. Speed is distance over time. Everyone knows that.

But is it?

Well first off, yes it is. The equation says so.1 But is that what it has to be?

Put yourself in the shoes of a caveman. A very smart caveman, mind you, but one that knows only rudimentary math.

After a couple hours watching tortoise v hare races, you want to really understand why the hare moves faster than the tortoise.

After a little pondering, you come to realize that the hare covers more distance in a smaller amount of time.

"Aha! I'll just divide the distance by—"

Woah there. Try not to fall into the trap of spitting back what you already know. Really see through the eyes of the caveman. How would you approach the problem if you didn’t have the machinery of middle school math?

You make some measurements. It takes .5 seconds for the hare to travel 3 meters.

Hmmmmm.

Well, if the hare travels .5 seconds through time and 3 meters through space on the same interval, they are, in some sense, equal. You write this down:

You know this next race will be 15 meters long, so you ask yourself: How many sets of .5 seconds will the race take? Well 15/3=5. Easy. The race will be 5 times longer than your measurement. Now 5 lots of .5 seconds is 2.5; 5*.5=2.5.2

That was relatively straightforward, but it’s a multi-step process. Multiple steps leave room for error, take longer, and are mathematically inelegant. It might be easier if you could condense it into one step.

So you want a single number that can help you determine how long any race will take. How do I find that number? Well look at the process we just went through. I divided the distance of the race by 3, then multiplied by .5. If I divide both sides of the previous equation by 3m, I can combine both of those processes in a single number:

Now you can predict how long it will take the hare to run any race simply by multiplying by the distance.

The next race is longer: 60 meters. Now how do we solve it? Multiply:

Notice what happened here. .5s/3m is a measurement of speed!3 But what's this? Time divided by distance? That's not right. Or is it?

The key realization here is that speed is not distance over time or time over distance. Rather, it's a relationship between the two.

Of course, in the real world we need have to have some standardization, so we settled on specific units: miles per hour or meters per second, but it's important to recognize speed for what it really is: a relation between time and distance.

1of course

2alternatively, you can just multiply both sides of the equation by 5, and get 2.5s = 15m, but that doesn’t move the story forward.

3Actually, in some circles (especially running), “speed” is thought of exclusively as distance over time, and “pace” as its inverse—time over distance. Others define “pace” as “the number of operations (eg footsteps or heartbeat) that occur per unit time” (as in “pacemaker”). Confusingly, this definition would give “pace” as distance over time.4 In this blog, I use the common notion of speed, which is “the rate at which someone or something is able to move or operate.” If you want your brain to dissolve into a heaping pile of semantic mush, you can read up on the various definitions at this link. (Note the multiple definitions of “speed” and the circular definition of “rate/speed”) You’re welcome for not including this in the blog.

4but maybe only for discrete notions of distance? “Operations” is a little vague, but usually doesn’t refer to continuous quantities.5

5Yes, my footnotes do have footnotes.

Noah CaplingerComment