Physics Of Racing Series Brian Beckman (2)

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.Let's find out how much time it takes to get through lines i and m.We include line o for completeness, even though it looks bad because it is both slower and longer than m.If we assume a maximum centripetal acceleration of 1.10 g, which is just within the capability of autocross tyres, we get the following speeds for the cornering phases of Lines i, o, and m:Cornering Speed (mph)Line i Line o Line m32.16 37.79 48.78vivovmLine m is all cornering, so we can easily calculate the time to drive it once we know the radius, labelled k in the figure.A geometrical analysis results ink = 3.414( Ro - 0.707 Ri) = 145 feetand the time isFor line i, we accelerate for a bit, brake until we reach 32.16 mph, corner at that speed, and then accelerate on the exit.Let's assume, to keep the comparison fair, that we have timing lights at the beginning and end of line m and that we can begin driving line i at 48.78 mph, the same speed that we can drive line m.Let us also assume that the car can accelerate at ½ g and brake at 1 g.Our driving plan for line i results in the following velocity profile:21Because we can begin by accelerating, we start beating line m a little.We have to brake hard to make the corner.Finally, although we accelerate on the exit, we don't quite come up to 48.78 mph, the exit speed for line m.But, we don't care about exit speed, only time through the corner.Using the velocity profile above, we can calculate the time for line i, call it ti, to be 4.08 seconds.Line i loses by 9/10ths of a second.It is a fair margin to lose an autocross by this much over a whole course, but this analysis shows we can lose it in just one typical corner! In this case, line i is a catastrophic mistake.Incidentally, line o takes 4.24 seconds = to.What if the corner were tighter or of greater radius? The following table shows some times for 30 foot wide corners of various radii:radius30.00 45.00 60.00 75.00 90.00 95.00to3.99 4.06 4.15 4.24 4.35 4.38ti3.94 3.94 4.00 4.08 4.17 4.21tm2.64 2.83 3.01 3.18 3.34 3.39margin1.30 1.11 1.01 0.90 0.83 0.82Line i never beats line m even though that as the radius increases, the margin of loss decreases.The trend is intuitive because corners of greater radius are also longer and the extra speed in line m over line i is less.The margin is greatest for tight corners because the width is a greater fraction of the length and the speed differential is greater.How about for various widths? The following table shows times for a 75 foot radius corner of several widths:width 10.00 30.00 50.00 70.00 90.00>to2.68 4.24 5.47 6.50 7.41ti2.62 4.08 5.32 6.45 7.51tm2.46 3.18 3.77 4.27 4.73margin 0.16 0.90 1.55 2.18 2.79The wider the course, the greater the margin of loss.This is, again, intuitive since on a wide course, line m is a really large circle through even a very tight corner.Note that line o becomes better than line i for wide courses.This is because the speed differential between lines o and i is very great for wide courses.The most notable fact is that line m 22beats line i by 0.16 seconds even on a course that is only four feet wider than the car!You really must "use up the whole course."So, the answer is, under the assumptions made, that the inside line is never better than the classic racing line.For the splice-type car behaviour assumed, I conjecture that no line is faster than line m.We have gone through a simplified kind of variational analysis.Variational analysis is used in all branches of physics, especially mechanics and optics.It is possible, in fact, to express all theories of physics, even the most arcane, in variational form, and many physicists find this form very appealing.It is also possible to use variational analysis to write a computer program that finds an approximately perfect line through a complete, realistic course.23The Physics of Racing,Part 6: Speed and HorsepowerBrian Beckmanphysicist and member ofNo Bucks Racing ClubP.O.Box 662Burbank, CA 91503©Copyright 1991The title of this month's article consists of two words dear to every racer's heart.This month, we do some "back of the envelope" calculations to investigate the basic physics of speed and horsepower (the "back of the envelope" style of calculating was covered in part 3 of this series) [ Pobierz całość w formacie PDF ]

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