Uh, that equation shows that acceleration is inversely proportional to mass…
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Pure 1st year Newtonian mechanics ignoring friction, air resistance, etc., he acceleration will be the same for both. Both will be going the same speed at the bottom of the hill.
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Actually, first year mechanics will include friction, and it would also have them going the same distance. However, first year mechanics will assume that friction is proportional to the normal force (for identical materials), which is not quite true. Close, but not quite.
You are correct that I should have said “the rate of acceleration of the respective kids downhill is dependent upon their respective masses, and the forces on them”.
You described my whole post as “wrong” over this one mis-phrasing (particularly given that based on the rest of the post it was clear I understood the correct position)?
OK, sorry, I guess I better understand what you’re saying now, and I do agree with you. I don’t think energy’s the best way to think about it though, because it’s not clear at all how the energy lost to friction or air resistance scales with size. You say it’s marginal, but it’s really not, especially for friction. The energy lost to friction scales almost linearly with mass. But I apologize for my earlier remark.
I tend to agree regarding an analysis in terms of energy. I find it much easier and more intuitive to talk about momentum and force etc (which is why I did that too) but IME someone in these threads always analyses in terms of energy, so I thought I’d get in first.
I dont have anything to add about the physics.
But I would like to say that the title of this thread deserves a prize-- for a unique combination of words.
Nobody in human history has ever used “fat kid” in the same sentence as “Galileo”.
I just wanted to add a real world example. In luge competition, riders are permitted to wear additional weight based on their initial starting weight. It’s a sliding scale in which riders who weigh less than a certain threshold are permitted to add ballast. The intent is to offset the benefit that heavier riders have over lighter ones. See here for an overview of the physics of luge.
First of all, empirically, a sled with a lot of weight on it definitely does go much further than a sled with little weight on it. The kids on top of the hill were correct: He went further because he’s fat.
But now we must figure out why that is. If the only force were gravity, mass would not matter. If the only forces were gravity and the normal force exerted by the snow, mass would still not matter. Even if we had gravity, normal force, and dry friction, mass would still not matter. This is because, in such a situation, all of the forces are directly proportional to the mass, and so the acceleration is the same for all (some fraction of g, with the fraction being determined by the steepness of the slope and the coefficient of friction).
If mass does matter (as, empirically, we know it does), then there must be some other force at work besides gravity, normal force, and dry friction. buddy431 has already pointed out one possible culprit: Friction with snow does not behave in the same way as dry friction, and so snow friction is not proportional to mass. Another possible culprit, which is more significant than many folks realize, is air resistance, which is also not proportional to mass.
If the kids were cub scouts they knew all about the forces involved from building their pinewood derby cars. Even among the youngest scouts you’ll see all the cars within .01 ounces of the legal maximum, center of gravity moved just in front of the rear axle of the car to maximize potential energy at the start, and axle nails buffed to a mirror finish to minimize friction. You weren’t going to fool them.
I was going to say this, along with soap box derby contestants and any toddler who’s had the pleasure of going down a water slide between his 200lb father’s legs. Shit gets real when you quintuple your mass on water slide.
Despite the advance physicism displayed in this thread I think the fat kid started further up on the hill and had the advantage of less friction on the snow surface from the skinny kids taking their runs. But Leo did the right thing, throw a bone to the fat kid.
They have a 5 oz weight limit on the cars for a reason. If they didn’t, the guy with the 20lb car would always beat the guy with the 10lb car; that is until someone’s 30lb car broke the track.
In the Pinewood derby, at any sledding hill, and implicitly stated in the OP (assuming that the sidewalk and the street have negligible sideways slant) there is a “steep” downhill portion and a flat (or flatter) runout portion.
You may choose to ignore friction in the downhill part, but you cannot on the flat part. Even if you assume all parties have the same speed and friction at the bottom of the hill, the more massive sled will have more momentum and kinetic energy, and therefore a longer runout.
I think that practically speaking, friction does increase for a heavier sled, but within the range of masses where friction increases at a slower rate than momentum, the heavies will win.
