4 ms·
Simple bouncing ball puzzle, with $50 prize
- deleted 16y ago[deleted]
- cperciva 16y agoThis is indeed a very nice puzzle.
- sswam 16y agothanks :)
- barrydahlberg 16y agoThe little boy picks it up and throws it again?
- cperciva 16y agoThere is no air resistance in this world, hence no air. The little boy asphyxiated shortly after the first time he dropped the ball. :-)
- barrydahlberg 16y agoHah! Let's hope it wasn't the little boy on his home page then... http://sam.ai.ki/ http://sam.ai.ki/
- TamDenholm 16y agoI'm no maths/physics geek so I'm just guessing here, but I'd assume that it loses energy at a far more rapid rate and maybe bounces once more and then just rolls along the ground. My reasoning is based on the fact that while the ball absorbs 40% of the energy on the higher bounces there is a minimum amount of energy it absorbs and on the smaller bounces the energy left meets the minimum amount of energy the ball will absorb and then simply stop bouncing. Like i said this is pure conjecture and I may not have even articulated it right.
- cperciva 16y agoGenerally things tend to get more elastic at low energies, not less; so I think your 'minimum amount of energy the ball will absorb' hypothesis is wrong. However, in the real world, there are other factors which drain energy from bouncing balls -- air resistance, for example. But this is all irrelevant, since the problem is explicitly in a non-physically-accurate world.
- burgerbrain 16y agoI'd love to know how many people have swamped him with the correct answer so far. Any junior high physic student could answer this.
- cperciva 16y agoI'm more curious to know how many people have sent in incorrect answers. My immediate guess disagreed with my mathematics.
- btilly 16y agoMy immediate guess agreed with my mathematics. But I have the advantage that a variant of this problem had occurred to me some 20 years ago, and I worked it out. In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely. There may be some residual vibration that is not apparent, but it sure seems to act like this toy model of the situation says it should.
- cperciva 16y agoIn fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely. Indeed, I've done that -- but my intuition told me that it was stopping due to friction, not due to the exponential decay of an infinite number of bounces. :-) It's oddly disappointing to realize that the model actually acts more or less the same as the real world for once.
- btilly 16y agoIndeed, I've done that -- but my intuition told me that it was stopping due to friction, not due to the exponential decay of an infinite number of bounces. :-) But your intuition was correct! Without friction the bounces would be perfectly elastic and wouldn't go into exponential decay! :-)
- 16y ago
- pero 16y agoWhere's the spoiler?
- btilly 16y agoHere is a spoiler. Energy = force * distance After each bounce you are left with 60% of its energy, so it comes back up 0.6 times as high as the previous bounce. Distance falling in time t is proportional to the square the time, so each bounce takes sqrt(0.6) times as long as the previous bounce did. Thus the timing of the bounces forms a geometric series. It is well known that the sum of such a geometric series is 1/(1-r). In this case r = sqrt(0.6) which is roughly 0.774596669241483 and so from the time it first hits the ground to the time it it finishes bouncing is approximately 4.43649167310371 times as long as the time for the first full bounce. But we didn't start with a full bounce, we dropped the ball. Thus we start with a half-bounce, followed by a full bounce that takes 2 * sqrt(0.6) times as long, followed by the rest of the sequence. This works out to be 7.87298334620742 times the time it took to initially fall to the ground the first time. Hopefully I haven't made any silly mistakes. If I have, correct the error and the general analysis is correct.
- greenlblue 16y agoSame mistake as everyone else. Geometric series means infinitely many bounces and infinitely many bounces means it never stops bouncing. Everyone is making the same logical fallacy.
- btilly 16y agoI refer you to Zeno's paradox for an example of how a geometric series can allow an infinite number of things to happen in a finite time. In this case the time taken forms a geometric series, and the total time taken is the sum of that geometric series. Which means that, for the same mathematical reasons that let Achilles catch the tortoise, it stops in finite time.
- greenlblue 16y ago
- greenlblue 16y agoSo I did some math starting with 100J of potential energy and summed a geometric series to see how long it would stay in the air if we had infinite time to sum the geometric series and it was some finite number. Then I assumed the initial velocity in the horizontal direction was 1 m/s so if we had infinite time the ball would travel some finite distance because it would bounce infinitely often and in every such bounce it would be in the air some amount of time and would travel at a speed of 1 m/s in that amount of time. But this assumes that the ball doesn't roll along the floor when it comes down and immediately bounces up as soon as it hits the floor which is unrealistic because it means I'm assuming infinite impulse on each bounce because the momentum of the ball changes and I'm assuming this takes 0 seconds. So the best I was able to do was give an upper bound on the total distance the ball would travel if the universe were to last forever. It's a lot like Zeno's arrow or is it Achilles and the tortoise.
- bravura 16y agoedit: I thought this over more, here is a new answer. Spoiler: it bounces an infinite amount of time in a finite space. Actually, the horizontal distance it travels is proportionately decreased with the decreased time in the air. If the ball is now in the air for k<1 times as much time, it goes to the right a factor of k times as much. I forget but I think these recursive series converge to a finite sum, they definitely do if k<0.5. So the height and length of each bounce decays exponentially, and it bounces an infinite amount of time in a finite space. --- Old, wrong answer. Spoiler: It keeps going to the right to infinity, just at a height that is vanishing and asymptotically approaching zero. Each time the ball bounces, it loses 40% of its vertical kinetic energy. But the problem statement ("the ground is flat, and each part of the ball’s path is a parabolic arc. Don’t consider friction, atoms, relativity, quantum, etc!") indicates that the horizontal energy doesn't change, even though the figure would suggest otherwise. So it will keep going to the right at the same rate.
- greenlblue 16y agoIt's true that the horizontal energy doesn't change and it keeps going to the right but it stays in the air for less and less on each bounce and travels a shorter and shorter distance on each bounce assuming the ball only goes forward when it bounces but you still didn't answer the question because if you sum the total time the ball will spend in the air you will get a finite number so the ball will travel a finite distance which is a much better description of what happens to the ball than just saying it keeps going to the right.
- mfukar 16y agoThe problem statement says 'ignore friction'. Doesn't that imply the ball's initial energy (when thrown, presumably, but it doesn't matter) is conserved? How, then, will it travel a finite distance, since we are not told of any obstacles?
- utoku 16y agoThe problems with questions in imagined worlds is that sometimes it is hard to get the premises right. Even if you treat the problem as a thought experiment, it feels uncomfortable because: - Ball bouncing with 60% energy remaining all the time is actually a premise, a rule, because we are not supposed to consider atoms or their interactions. There is no reason to it, it becomes a fact. - Ball is a singular object. We don't have atoms, so we don't have to think things like "What happens when the height of the bounce gets shorter than the size of the atom?" The concept of the ball becomes a premise. - We get a picture that shows the balls losing speed in direction x at each bounce, but since the question asks for us to judge "qualitatively", we will omit that. We cannot have observations for this problem, it is not the real world. - There is no friction, or spin, so vertical speed cannot be transfered into horizontal speed and vice versa. - The concept of bounce might be different, this question probably assumes it happens at 0 (instant) time without deforming the ball, since remember our ball is singular. - The rest we can probably treat with Newtonian physics in Euclidean geometry, no air, interaction between ball and surface frictionless, etc. But it feels uncomfortable, because I too can make up an imaginary world for myself, dress it like the real world and ask my question and hide the premises behind.
- deleted 16y ago[deleted]
- retube 16y agoThe ball has horizontal energy as implied by the parabolic arc. Assuming no friction the ball's horizontal velocity will be constant, hence it will just roll away at whatever initial horizontal velocity it had.
- hardy263 16y agoI love how there's so many calculations with energy and time. Here's my offering with just concept. The first thing they teach you in projectile motion in grade 12 physics is that the horizontal component is independent from vertical component. That means, if you ignore friction, and you throw a ball in an arc, you'll find that the horizontal speed is linear! This surprises many people, since it's not very intuitive. You would expect the horizontal speed to be quadratic or non-linear, which is not the case. If the ball loses 40% of its vertical energy, it means that it'll just keep bouncing, but at lower and lower heights, but the horizontal speed is continuous. In fact, if given the initial velocity and angle, you could calculate the horizontal distance traveled by the ball for any point in time.
- drallison 16y agoAny horizontal component to the motion can be ignored. The only thing of interest is the vertical motion of the ball in a gravitational field. There is an elastic collision between the ball and the flat surface. Unlike an inelastic collision, energy is not conserved. Energy is lost when the ball hits the surface and rebounds; it shows up as heat in the ball and at the surface at the point of collision. Assuming the surface is very hard, most of the heat goes into the ball. At some point the kinetic energy remaining is not enough to lift the ball against gravity so the ball does not get lifted off the surface. If you follow the center of mass of the ball, it continues to oscillate, compressing and expanding elastically until the remaining kinetic energy is expended as heat. As the size of the oscillations get smaller and smaller you eventually reach a scale where the idealized model of the ball begins to fail; at that point, things become complicated.
- deleted 16y ago[deleted]
- sswam 16y agoCongratulations to everyone who figured it out, and especially to Glenn from Alaska who was the first to email me with the correct answer, and wins $50. In his words: "The ball makes an infinite number of progressively smaller bounces in a finite amount of time, and then proceeds to slide (roll?) along the ground at a constant speed." I was just looking for "infinite number of bounces in a finite amount of time/distance". I think it's a nice puzzle, because it illustrates Zeno's 'paradox', with a simple model of an everyday occurrence. The answer makes sense, but it is not obvious unless you understand limits. I thought of this puzzle while playing with a pool cue, you can really hear/feel them bouncing faster and faster.