## How Fast Could They Have Gone?

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I love the show Top Gear. In particular, I am a big fan of the “challenges” they have. Recently (yesterday afternoon, when I should have been grading), there was an episode on BBC America from 2006 in which host Richard Hammond was at London’s ExCeL Centre for the British Motor Show, and before the exhibits were set up, he had The Stig (the show’s “tame racing driver”) test two vehicles to see how fast they could go in the 385-metre hall. The first was a Chevy Lacetti, their “Star in a Reasonably-Priced Car” car, which reached 70 mph. The second was a Toyota F1 car (the TF105, I think), which reached…

only 81 mph?

This led me to wonder what the top speed of a car could be on a 385-metre stretch. Let’s find out.

For simplicity, I will assume constant acceleration a (I said simplicity, not accuracy), and constant deceleration from braking (again, not particularly realistic). Let $v_T$ be the top speed. Then the distance covered in accelerating to top speed is

$\displaystyle\frac{1}{2}a \left( \frac{v_T}{a} \right)^2$

and the braking distance is

$\displaystyle \frac{v_T^2}{2\mu g},$

where μ is the coefficient of friction between the tires and the floor, and g is gravity. We then have the following equation:

$\displaystyle \frac{v_T^2}{2} \left( \frac{1}{a} + \frac{1}{\mu g} \right) = 385$

and thus

$v_T = \displaystyle \sqrt{\frac{770a\mu g}{\mu g+a}}.$

At this point, I plead ignorance. I tried (not very hard) to find a reasonable coefficient of friction for racing tires, and to find a 0-60 time for a Formula 1 car (the McLaren F1 does it in 3.9 seconds). In the end, I made a spreadsheet for μ between 0.4 and 0.7, and a between 6 and 7 m/s2. Given an ideal setup—starting at one end of the hall and stopping perfectly at the other end—the top speed is somewhere between 95.6 mph (μ=0.4, a=6) and 115.5 mph (μ=0.7, a=7).

What does this mean? I don’t know, but it makes The Stig’s 81 mph sound pretty good, given the initial burnout, the nonconstant acceleration and braking, driver reaction time, and an interest in personal safety (cf. Hammond’s “Formula 1 car-shaped hole” comment).