Calculating Maximum Speed of a Bike Downhill

[tycon69]

Well, let's say we have a bike and we wanted to see the maximum speed that the bike would reach down a hill. Let's say that the combined weight of the rider and the bike is 210 lbs. The diameter of the wheels is about 20 inches and the hill is 45 degrees steep. The hill is smooth and has just enough surface tension to hold the bike up. The hill extends infinitively or however long it needs to be for the bike to reach maximum speed at these conditions. I was wondering if this Maximum Speed could be determined mathematically. I have yet to take a physics course, so i really know nothing of mechanics or the such, and i was wondering if someone could show me how we would calculate such a problem.

 

[russ_watters]

Under such conditions, the maximum speed is a terminal velocity determined by air resistance.

 

[rcgldr]

With an aerodynamic riding suit, unfaired bicycle downhill record on snow is 132mph, on dirt (actually lava bed from volcano), 107mph (bike broke, but rider survived). I don't know how steep the hills were though.

 

[Hootenanny]

Just a conceptual comment, the bike will never actually reach its terminal (maximal velocity) in a finite time, instead the bike's velocity will come arbitrarily close to the terminal velocity. If we assume rectilinear motion (the bike travels in a straight line down the hill), then finding the terminal velocity involves solving an ODE of the form,

$$m\ddot{x} = mg\sin\theta - k\left(\dot{x}\right)^2 = 0$$

Which has quite nice solutions (assuming the density of air remains approximately constant). To determine the value of k, we would need information on both the cross-sectional area of the bike/rider and the numerical drag coefficient (which we could approximate). If you like I could detail the solution here, but I'm not sure how useful it would be if you haven't done any calculus before.