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If you drive the tire onto a rail half as wide as the width of the tire (specifically, the width of the area in contact with flat ground), then according to his logic, the length of rail in contact with the tire will be twice the length of flat ground in contact with the tire (assuming the contact area to be roughly rectangular in each case). I have no doubt that the length would be more, but severely doubt it would be anything close to twice as much because I think a large percentage of the "normal force" (the weight of the car) goes into keeping the tire bent out of shape in ways that do not contribute linearly to the area in contact with the rail.

My guess is that the tire's resistance to being bent out of shape is also a significant factor in the usual case of the tire's resting on flat ground and that he got as close as he did through a combination of coincidence and cherry picking (deciding after he knew the right answer how much the added gas weighs, changing his estimate of how much of tire's surface consists of grooves after he knew the right answer).

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