On Bicycles, and.... what else is there?

I came across the following interesting tidbit on the CyclingNews review of the Trek Speed Concept 9 : To tweak the original Speed Concept, Trek engineers studied real-world wind conditions on Ironman courses, and settled on optimizing the bike for yaw angles between zero and 12.5 degrees. (They found 3-5 degree average yaw in Arizona, up to a 13 degree average yaw in the notoriously windy Kona worlds course.) I came up with my own ad hoc calculation, including the Hellman model for ground shear, back in 2011. I published that here : So the end result of all of this is that, given the assumption of an average 10 mph wind at 10 meter elevation (a stiff wind), a Maxwellan distribution of the ratio of wind speed to rider speed, an average rider speed of 11 m/sec (40 kph), and a Hellman coefficient of 0.34 (associated with urbanized areas), I get the average yaw angle magnitude is 6.8 degrees, with yaw probabilities extending from -20 degrees to +20 degrees. The yaw of greatest probab...

Last time I gave up trying to solve this integral. I wanted to watch Cyclocross Worlds so got lazy and did a numerical solution and fit. But I realized while out running after the racing that I could have done much better. So I'm back for more. (1 / 2π) ∫ dφ exp[ (s w ' / 3) 2 ] exp[ −2 s wx ' / 3 ], where the integral is over the full circle and φ is the angle of the wind relative to the rider (0 = pure tail wind). The key here is to recognize that this can be well-approximated by a Gaussian for s w ' to at least 1. This isn't the solution of the integral, but it's a good approximation, so once I recognize the analytic form of the solution I can get away just matching derivatives with respect to s w ' = the ratio of the wind speed to the zero-wind rider speed. So my solution will be the following, where I must solve for K: exp[ (s w ' / K) 2 ]. I recognize that for every value of positive s wx ', there is a corresponding negative value...