Hercules and the Tortoise

One of the most frustrating maths questions you’re likely to get involves relative speeds. It’s not that the concepts are hard – it’s just that you probably never got taught the concepts at all! And relative speed is one area where conceptualisation makes all the difference.

To see what I mean, consider the ancient Greek philosopher, Zeno. Zeno was different to all the other ancient Greek philosophers, in that his goal wasn’t to show that logic and rationality could lead you to understand the universe – it was to prove that logic and rationality simply aren’t enough!

To show this, he developed some paradoxes – conclusions, logically arrived at, that are totally absurd. One of his most famous involved Hercules and the tortoise. The story goes like this:

Hercules is Big and Strong and Fast. The turtle is … not. So, if Hercules were to chase a turtle, you gotta figure he’s gonna catch him, right?

Hold on, says Zeno. Hercules would have to run up to the place where the turtle was to start with – and during that time, the turtle would have moved a bit. So, Hercules would have to run up to where the turtle is then – during which time, the turtle will have moved a bit. So, Hercules would have to run up to where the turtle is now – during which time, the turtle will have moved a bit. And that will keep happening … forever.

So, logically, Hercules never catches the turtle – even though he’s Bigger and Stronger and Faster!

Now, obviously, Hercules will absolutely catch that turtle in reality – the point is that, proceeding perfectly logically, we’ve ended up believing something crazy.

So, does this mean maths are insane and we should chuck it all in to live on a commune? Not necessarily (unless you’re into that kind of thing, anyway). Instead, recognise this: it doesn’t matter how fast Hercules is – it matter how much faster he is than the turtle!

Let’s say Hercules can run 12 mph, and the turtle can only run 3 mph. Hercules is currently 27 miles behind the turtle. How long will it take to catch him? Instead of trying to create simultaneous equations and thinking about two bodies in motion, let’s just say this: Hercules is 27 miles away from the turtle and he’s catching up at 12 – 3 = 9 miles every hour. So, at that rate, how long would it take Hercules to catch up? 27/9 = 3 hours. Simple as that.

After all – how fast are you moving right now? You might say zero miles per hour – but you are on the Earth and the Earth is rotating, no? And the Earth is circling the Sun, right And the Sun is circling the Galaxy, the Galaxy is circling the Universe and the Universe is accelerating in all directions at once! So how fast are you really going? Well, zero – compared to the Earth!

All speed is relative speed, just as all distance is relative distance. And there’s no reason to privilege Hercules’ speed compared to the Earth, when what matters is the turtle!

Simply conceptualising the problem this way makes an otherwise complicated issue incredibly easy. But then, that’s the GMAT’s way, isn’t it?

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