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Playing with infinity: the basketball, part 2 (part 1 solved)


Best Answer Yoruichi-san, 20 June 2013 - 12:34 AM

Spoiler for Since no one else is answering...

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11 replies to this topic

#1 bonanova

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Posted 12 June 2013 - 05:57 AM

An idealized basketball falling from a height h bounces from the floor to a height h/2.

Tell us two things:

  1. The ball is dropped from a height of 1m.
    Does it come to rest (stop bouncing) in finite time?

     
  2. Xavier, in post #5 shows us that after 9.31 seconds, the ball comes to rest.

    We now specify that the ball is
    blue initially and on each bounce it changes color,
    alternating between
    blue and red. After the ball comes to rest, it ceases to change color.


    Question 2: what is its color after coming to rest?

Edited by bonanova, 13 June 2013 - 07:59 AM.
Pose Part 2.

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#2 Barcallica

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Posted 12 June 2013 - 09:04 AM

Spoiler for


 


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#3 James33

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Posted 12 June 2013 - 09:57 AM

Spoiler for


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#4 Quantum.Mechanic

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Posted 12 June 2013 - 11:38 AM

Please define "ideal". To me, "ideal" means the ball would bounce back to height h (assuming various things like the floor is the frame of non-rotating reference, perfectly elastic collision, etc.)

 

Since you state the ball bounces back to h/2, something else must be going on. For instance, the ball at  h  could be going faster than terminal velocity in air a . Or the floor is not a perfectly elastic surface.


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#5 Xavier

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Posted 12 June 2013 - 10:11 PM

Assuming the Ball instantly loses half its velocity when it hit the ground: 

Spoiler for Hint

 

Spoiler for Answer

Edited by Xavier, 12 June 2013 - 10:14 PM.

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#6 bonanova

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Posted 13 June 2013 - 03:14 AM

Please define "ideal". To me, "ideal" means the ball would bounce back to height h (assuming various things like the floor is the frame of non-rotating reference, perfectly elastic collision, etc.)
 
Since you state the ball bounces back to h/2, something else must be going on. For instance, the ball at  h  could be going faster than terminal velocity in air a . Or the floor is not a perfectly elastic surface.

 

By elastic I would mean returning to the full height h. This ball is not elastic.

By ideal I mean to say neglect air resistance, energy loss to acoustic processes, energy lost to deformation of the ball that might depend on impact velocity, and so on. That is, this idealized ball has none of these bothersome second-order details; it has a behavior that is completely described for the purpose of the puzzle as "returning to half the height, on each bounce, from which it fell."

Aside from this, the laws of physics, in particular a constant acceleration due to gravity, apply.


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#7 bonanova

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Posted 13 June 2013 - 03:30 AM

Assuming the Ball instantly loses half its velocity when it hit the ground: 

Spoiler for Hint

 
Spoiler for Answer

Zeno indeed is watching this ball.
Not every infinite series converges, but the infinite sequence of bounce times in this case does sum to a finite number.

According to Xavier, 9.31 seconds after the ball is dropped, it comes to rest.
Part 1 of the basketball puzzle is solved.

The OP has been edited to include Part 2.


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#8 James33

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Posted 13 June 2013 - 02:06 PM

 

An idealized basketball falling from a height h bounces from the floor to a height h/2.

Tell us two things:

  1. The ball is dropped from a height of 1m.
    Does it come to rest (stop bouncing) in finite time?

     
  2. Xavier, in post #5 shows us that after 9.31 seconds, the ball comes to rest.

    We now specify that the ball is
    blue initially and on each bounce it changes color,
    alternating between
    blue and red. After the ball comes to rest, it ceases to change color.


    Question 2: what is its color after coming to rest?

 

 

Spoiler for


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#9 bonanova

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Posted 13 June 2013 - 08:20 PM

An idealized basketball falling from a height h bounces from the floor to a height h/2.Tell us two things:

  • The ball is dropped from a height of 1m.
    Does it come to rest (stop bouncing) in finite time?

     
  • Xavier, in post #5 shows us that after 9.31 seconds, the ball comes to rest.

    We now specify that the ball is
    blue initially and on each bounce it changes color,
    alternating between
    blue and red. After the ball comes to rest, it ceases to change color.


    Question 2: what is its color after coming to rest?
 
Spoiler for
And after it stops bouncing?
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#10 TimeSpaceLightForce

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Posted 14 June 2013 - 02:41 PM

Spoiler for not infinite


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