The Area Under A Distance-Time Graph

Teaching Mechanics

I love teaching A level maths, but I must admit to feeling a bit disappointed when I found out I would be teaching the Mechanics module (M1 from MEI, fact fans) because... Should I really be saying this? OK, I'll just say it: I find it boring. No, that's not the whole truth, I'm making excuses now. I have to admit: I don't always get it.

I did Mechanics 1 at A-level (also using the MEI exam board) and that's the full extent of my knowledge. No Mechanics 2, none at Uni, I didn't do Physics A level or even separate Physics GCSE. I've never really been a "scientist", preferring to see myself as a creative type. I was always better at English and French than at science.

Anyway, this year (2012) I've been team-teaching M1, sort of. I've basically been watching someone else teach it. And suddenly I'm learning things I never really understood the first time round. I'm starting to get it!

The Area Under a Distance Time Graph

But then I went and spoiled it all by doing something stupid like Googling "What's the area under a distance-time graph?"

I would hope that those reading this are aware of the concepts illustrated below:

When you draw a velocity-time graph, for example, the area underneath the curve gives you the displacement, and the gradient of the curve gives you the acceleration.

The other teacher of the class posed the question to the class "What's the area underneath a displacement-time graph?". My mind was immediately blown. With a bit of jotting down (the diagram above), I could quite easily work out that the units had to be ms (metre seconds, a bit like kilowatt hours), so I knew it was something to do with distance multiplied by time.

I knew that what I was looking for would be the blank in this sentence: "displacement is the rate at which BLANK changes". What could fit? Nothing seemed intuitive.

Absement

That was where that much-relied on search engine came in. After a bit of research (Googling), I found that the word I was looking for was "absement" (oh of course, you all exclaim) and that I was by no means the only geek in the world wondering what it was.

Absement is a port-manteau of the words absent and displacement (knowing which makes it no easier to understand) and there are not many results on Google for it (the eighth result is an English to Urdu translation page. Erm, cheers for that).

I will attempt to explain absement using an example. For another example, visit Wearcam.

You live 2km from school. You walk to school in 30 minutes, stay there for six hours, then return home, also taking 30 minutes.

A displacement-time graph would look like this:

Now let's consider the area under this graph. It would be given by integrating the curve above. You can sketch this curve easily: in the first section, the gradient is constant and positive, so the corresponding bit of our new graph would be increasingly increasing (curving upwards). The middle bit has zero gradient so our new graph will have a constant positive gradient. The last bit is the opposite of the first bit, so our new graph's last section will have a decreasingly increasing bit.

My sketch:

The next thing to do is calculate the numbers on the vertical axis.

Your absement at any point is given by the (average) distance you are from home multiplied by the time that you're there for.

So when you've just arrived at school, your absement for that point is 1000m (your average distance from home) multiplied by 1800. So that's 1 800 000 ms. Up to that point, your absement has been increasingly increasing. When you're halfway to school, for example, your average displacement was 500m, your time is 900, so your absement is 450 000ms (note that this is not half of 1 800 00ms). At the end of your second part (when you've just finished school), your displacement has been constant at 2000m, your time has been 6 hours which is 21600 seconds, so your absement at that point is 43 200 000 ms. You can calculate your absement at many different points so that you can plot a nice, smooth curve. It looks a bit like a cumulative frequency curve. I'll leave this to you as an exercise.

What's the point of absement?

Well if you were on a spaceship and you had some kind of mobile communication device, you might imagine that the further from Earth you are, the more power the device uses. Therefore you might measure its battery usage in metre seconds.



Emma x x x