So 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.
Simple bouncing ball puzzle, with $50 prize
11–20 of 58 posts
Re: Simple bouncing ball puzzle, with $50 prize
#12edit: 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 kit bounces an infinite amount of time in a finite space.
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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.
Re: Simple bouncing ball puzzle, with $50 prize
#13edit: 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 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…
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 distance which is a much better description of what happens to the ball than just saying it keeps going to the right.
Re: Simple bouncing ball puzzle, with $50 prize
#14edit: 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 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…
Spoiler: That was my first guess.
Re: Simple bouncing ball puzzle, with $50 prize
#15edit: 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 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…
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.
Re: Simple bouncing ball puzzle, with $50 prize
#16edit: 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 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…
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 potential energy then you know how high the ball is and if you know how high the ball is then you know how long it will take for it to fall to the ground but no matter how long you wait the ball will have some potential energy left so it will be bouncing no matter how long you wait.
Re: Simple bouncing ball puzzle, with $50 prize
#17[deleted]
Re: Simple bouncing ball puzzle, with $50 prize
#18Earlier quoted context omitted.
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…
The bounces are an infinite series, each with a time attached to them tending to zero.
Time itself is not a series, this is where the fallacy creeps in.
A infinite series can have a finite sum.
Re: Simple bouncing ball puzzle, with $50 prize
#19The little boy picks it up and throws it again?
There is no air resistance in this world, hence no air. The little boy asphyxiated shortly after the first time he dropped the ball. :-)
Hah! Let's hope it wasn't the little boy on his home page then...
http://sam.ai.ki/
Re: Simple bouncing ball puzzle, with $50 prize
#20edit: 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 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…
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