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        riddles >> medium >> Wagon Wheel
        
(Message started by: ThudanBlunder on Jul 8^th, 2008, 3:22am)

Title: Wagon Wheel
Post by ThudanBlunder on Jul 8^th, 2008, 3:22am

A wagon wheel of radius R is thickly coated with mud and the rim, which moves at a constant speed of v, is continually throwing off mud from every point. Can you help the passengers find out what is the maximum height above the ground reached by the mud?

Title: Re: Wagon Wheel
Post by towr on Jul 8^th, 2008, 4:32am

I seem to have a bit of a problem with low speeds, but
[hide]
The height for the radial angle at which the mud flies off is
R + R sin(t) + 1/2 cos²(t) v²/g
Differentiating, I get
R cos(t) - cos(t)sin(t) v²/g
So we have optima at
cos(t)=0 and Rg/ v^2 = sin(t)
So either the maximum is 2R, or
R + R² g / v² + cos²(arcsin(Rg/v²)) v²/(2g) =
R + R²g/v² + (1-(Rg/v²)²) v²/(2g) =
R + R²g/v² + (1-R²g²/v⁴) v²/(2g) =
R + v²/(2g) + 1/2 R²g/v² =
(gR + v²)²/(2gv²)

Of course, I've got a bit of a problem there with height going to infinity when speed goes to zero. But the other end of the curve works out, I think.
[/hide]

Title: Re: Wagon Wheel
Post by Eigenray on Jul 8^th, 2008, 5:23am

The behavior changes when [hide]v² < gR[/hide]: you're never going to do any better than 2R. But otherwise, yeah, what towr said. Note that the best mud reaches exactly g/(2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/omega.gif²) higher than the mud that flies off vertically.