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The 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.
just roll away Technically, it will slide , since without friction it can't obtain any angular momentum.
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I'm not disagreeing with you. Yes, an infinite series can have a finite sum, no argument there but what I am arguing with is that people are saying the ball stops bouncing after time t = whatever. Everyone is confounding two things here, air time and bouncing. Yes, the air time is finite but the bouncing isn't. So you can't say it stops bouncing after time t = whatever if you calculate t = whatever by assuming infini…
There is a time t such that after time t the height of the ball's bounces is zero. Personally, I call that "not bouncing".
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It'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 d…
The 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?
- 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.
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There is a time t such that after time t the height of the ball's bounces is zero. Personally, I call that "not bouncing".
Ok, now we are in agreement because my definition of "stops bouncing" was finitely many bounces.
If you have infinite bounces, and I ask you "Which number bounce in the series happens at exactly time X?", there is a time for X for which you will not be able to give an answer.
This is because there is a limit for the latest time at which bounces happen.
Yes, even with infinite bounces.
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The 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?
The 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. W…
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Ok, now we are in agreement because my definition of "stops bouncing" was finitely many bounces.
Bzzt Still wrong. If you have infinite bounces, and I ask you "Which number bounce in the series happens at exactly time X?", there is a time for X for which you will not be able to give an answer. This is because there is a limit for the latest time at which bounces happen. Yes, even with infinite bounces.
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.
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My 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…
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. 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 t…
But your intuition was correct! Without friction the bounces would be perfectly elastic and wouldn't go into exponential decay! :-)
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This assumes that each bounce takes the same time to complete instead of also tending to zero. If you expect that each bounce also takes a smaller amount of time to complete, a series of infinite bounces takes a finite amount of time and therefore a finite distance. At the point in time and distance where the bouncing ends, the forward motion of the ball (unimpeded by friction) continues in a slide along the ground.
Ah but it takes infinite amount of time for the ball to stop and start rolling on the floor and infinite amount of time means the ball bounces infinitely often so the ball never stops to bounce and never rolls along the ground. It is easy to calculate how long the ball stays in the air on each bounce a formula from high school physics tells you that the potential energy of an object is m x g x h so if you know the po…
Unless of course it was rolling in the first place and I missed/misread it.
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.