## Wednesday, May 06, 2009

### Marcus du Sautoy and Formula 1

Marcus du Sautoy, the Charles Simonyi Professor for the Public Understanding of Science at Oxford University, makes an attempt in The Times to find a mathematical perspective on Formula 1. He begins:

This year Formula One teams that agree to race within a strict budget rather than relying on unlimited funds to develop their cars are being rewarded with a number of perks under a scheme called the Kinetic Energy Recovery System [KERS].

Marcus seems to have got his wires slightly crossed here: the opportunity to exploit greater technical freedom in exchange for a voluntary budget cap, does not arrive until 2010, whilst KERS is already available to anyone this year, irrespective of budget. Not an auspicious start then.

However, Marcus then redeems himself by posing an interesting question which does, by analogy, help to explain why the teams with double-decker diffusers have done so much better this year than the teams with KERS alone:

You arrive at an airport for a flight connection but the timing is tight. You discover that your next plane is leaving from the far end of the airport. There is a moving walkway for part of the journey. You’ve got enough energy to do a short burst of running, but otherwise you’ll walk at a constant speed...You want to get to the gate as quickly as possible, so the question is: when should you use your burst of energy? Should you run on or off the moving walkway?

The answer is that the button should be hit off the walkway:

Our twins are walking together towards the walkway when one twin decides to press the boost button to get him to the walkway before his brother. He reaches it D metres ahead of his brother. But as soon as he steps on to the walkway the distance between them begins to increase even more. When the second twin steps on to the walkway he presses his booster. Since both are on the walkway, this will allow him to catch up only D metres on his brother. But his brother is more than D metres away by now.

Marcus thinks that an analogy can be made with the question of whether the KERS boost button should be used on the slow part or the fast part of a circuit, but in practice, it can only be used to benefit on the straights; if used in the corners it would simply result in wheelspin.

The airport scenario does, however, still provide a good F1 analogy in the following sense: more time is spent, per unit distance covered, off the walkway, hence the greatest benefit is gained by hitting the boost button there. Similarly, in Formula 1, more time is spent, per unit distance covered, in the corners than on the straights, hence in terms of lap-time alone, the greatest benefits accrue from improving the cornering characteristics of the cars than by improving straightline speed. As a fresh demonstration of this, the teams which have developed a cornering advantage this year (the double-decker diffuser teams), are significantly faster than those which initially developed only a straightline speed advantage (the KERS teams).

Bob said...

Math. I can do that.
Walking speed is 5 kmh. Boost speed is 10 kmh. The speed of the moving walkway is 10 kmh. Assuming you have to walk 2 km normally and 2 km on the moving walkway. The energ for the boost lasts 5 minutes.

I won't tire you with the calcualtions, but you're right. The slow booster covers the complete distance in 22 minutes, whereas the fast booster needs 30 minutes and 20 seconds.

Strange. Intuitively I would have said that it doesn't matter.

That means that on my way to work, it is more important to drive fast in the city then it is on the highway. Alas, the pedestrians and the cyclists might not agree.

Gordon McCabe said...

My first reaction was also that it shouldn't matter.

On the latter point, note also that if you travel faster in residential areas, then you spend less time in those areas. Hence, if the probability of hitting a pedestrian or cyclist per unit time is independent of your speed, then you will minimise your chances of such a collision by travelling as fast as you possibly can!

From the perspective of the cyclists and pedestrians, their chances of getting hit by at least one car don't change, and when they do get hit, they get hit harder, but from the selfish perspective of the individual motorist, it makes sense to hurtle through built-up areas as fast as possible!

Sean said...

F1 has always been about the cars and drivers who can get around corners the fastest.So no change there.

If there much performance difference between the mechanical and electronic Kers system?

Gordon McCabe said...

I don't think we've seen Williams use their flywheel system at a race yet, so the jury is out on that one.