Answer to the Question 04/00

BIKE RIDE

The question was:



Suppose your bike has no rear fender. How slowly do you have to ride on a wet road to avoid getting mud on your bottom?


(7/2000) The problem has been solved by Jared D. Kaplan, - a high school student from the Illinois Mathematics and Science Academy (e-mail jared@imsa.edu). Partial answers have been also provided by Avi Nagar (e-mail anagar@bigyellow.com), by Regis Lachaume, a PhD student at Grenoble Observatory (France) (e-mail Regis.Lachaume@obs.ujf-grenoble.fr), and by Ido Golding, a PhD student at Tel Aviv University (Israel) (e-mail golding@gina.tau.ac.il).

The answer: About 12 km/hour (3.3 m/sec).

The solution: Below we present a (slightly edited) version of the solution submitted by Kaplan.



For this problem I set up a coordinate system, where the center of the rear wheel is at the origin, the back of the bike seat is at (d,h), and the radius of the rear wheel is r. The way I worked the problem was to express the height of a mud particle at a distance d from the origin in terms of its velocity upon leaving the wheel. The only tricky thing was dealing with what point on the wheel the mud leaves from. I expressed this as the angle A that the radius to the point of departure makes with the negative x-axis in my coordinate diagram, knowing the mud would be flying off perpendicular to this radius. Also, let the velocity of the bike be v:

vx = v*sin(A)
vy = v*cos(A)

Let the function H(t) denote the height of the mud particle at time t, and let D(t) denote the horizontal distance covered at time t:

H(t) = v*cos(A)*t - g/2*t2 + r*sin(A)
D(t) = v*sin(A)*t - r*cos(A)

So when D(t) = d, if H(t) > h, you get mud on your back.
D(t) = d --> v*sin(A)*t - r*cos(A) = d --> t = (d+r*cos(A))/(v*sin(A))

Plugging this time into our H function we get:

H(d/v*csc(A) + r/v*cot(A)) > h
cot(A)*(d+r*cos(A)) - g/2*(d/v*csc(A) + r/v*cot(A))2 + r*sin(A) > h

we can solve it for v and get:

v^2 < g(d+r*cos(A))2/(d*sin(2A)+2r*sin(A)-2h*sin(A)2)
v < sqrt(g/(d*sin(2A)+2r*sin(A)-2h*sin(A)2))*(d + r*cos(A))

What this all means is that if the velocity is less than this function at all A's, then we won't get dirty. I don't think this function can be minimized in general, but I applied it to my particular bicycle with measurements of:
h = .58 meters
d = .20 meters
r = .33 meters

I found numerically the minimum point of the above function to be approximately 3.28 m/s, meaning that as long as the velocity that the mud splatters off at is less than this, I won't get dirty. This is really nice because the this velocity is exactly the same as the maximum velocity of the bike. I didn't take drag with the air or any other more subtle features into account, but I hope this was at least a good approximation.

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