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Topic: Wagon Wheel (Read 521 times) |
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ThudnBlunder
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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?
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towr
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Re: Wagon Wheel
« Reply #1 on: Jul 8th, 2008, 4:32am » |
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I seem to have a bit of a problem with low speeds, but
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.
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| « Last Edit: Jul 8th, 2008, 4:36am by towr » |
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Eigenray
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Re: Wagon Wheel
« Reply #2 on: Jul 8th, 2008, 5:23am » |
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The behavior changes when v2 < gR: you're never going to do any better than 2R. But otherwise, yeah, what towr said. Note that the best mud reaches exactly g/(2 2) higher than the mud that flies off vertically.
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| « Last Edit: Jul 8th, 2008, 5:31am by Eigenray » |
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