A little classical mechanics problem you can solve without doing any calculation:
Consider the hyper-simplified problem of a bell-shaped hill, and a point rock that can slide without friction up and down the hill. If you start with the rock at the bottom, and give it exactly the kinetic energy needed to arrive to the top and stop there without sliding on the other side, how long will it take to arrive there?
#Physics #ITeachPhysics
@j_bertolotti - A looooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooong time. 🙃
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@johncarlosbaez @j_bertolotti I can't see how this can be deduced without some fairly sophisticated argument. If you project a particle straight up, it gets to the top of its trajectory in finite time. I'm pretty sure that if the surface came to a point (pointing upward) this would still be true, so it seems to rely on the surface being sufficiently smooth, so that the rate at which the particle slows on its way up means that it doesn't make all the distance in finite time. But I can't think of a convincing calculation free argument for this.
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@RobJLow
As Newton laws are symmetric with respect to time reversal, the time needed to get up must be equal to the time needed to go down. But if you start at the summit of a bell-shaped curve (which is horizontal) with zero velocity, it will take you infinite time to get down 😉
@johncarlosbaez
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@j_bertolotti @johncarlosbaez @RobJLow but this also mean that after infinite time, at some point the ball will start to move infinitesimally slow and than accelerate. And finally slide down the hill and far away
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@j_bertolotti @KodeGhinn @johncarlosbaez You have no patience.
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@RobJLow @j_bertolotti @KodeGhinn - normal mathematicians would say it's meaningless to talk about what happens to the ball after an infinite amount of time has passed, but if we replace the real numbers but the nonstandard reals or the "long line" it might make sense. It's not important in real-world physics, but I can definitely imagine mathematicians trying to use these frameworks to study what happens in a physical system "after an infinite amount of time has passed". Tell us something makes no sense and we'll try to figure out a way it can! 💪
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@johncarlosbaez @j_bertolotti @KodeGhinn Yes, you could consider how long it takes the ball to get closer an infinitesimal distance to the tip and fall back, and the answer would be a hyperfinite time: and this would be (could be could be interpreted as) describing the rate at which the time taken goes to infinity as the distance to the summit tends to zero.
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@KodeGhinn
What happens after an infinite amount of time is of no consequence. 😉
@johncarlosbaez @RobJLow