A perfect bouncing ball in a lossless system ...

[TaoRich]

Hi,

 

Something I pretty sure about was thrown into doubt this evening by some opinions outside this forum. I'm hoping someone can put me back on track and into my old comfort zone.

 

What is the behaviour describing a perfect bouncing ball in a lossless system ...

 

Now I know it's not true Simple Harmonic Motion (SHM) since we have at one end:

• the bounce
  • a high-speed drop reaching maximum velocity
  • an 'instantaneous' reversal of direction
  • a high-speed climb starting at maximum velocity

  [*]the apex

  • a slowing of the rate of climb from maximum velocity to 0
  • an 'instantaneous' reversal of direction
  • an accelerating rate of fall from 0 to maximum velocity

So we definitely do not have a complete sinusoidal wave

 

This is not what we have:

 

 

But am I correct in assuming this is what we have:

 

 

Half a sine wave, repeated.

 

Essentially, to my mind, isn't the wave motion reflecting about the X axis, just as the ball is reflecting about the point of its bounce impact.

 

Whereas in true SHM, the wave motion passes through the X axis, just as a pendulum would pass through the trough point of its swing.

 

Comments ?

Edited February 20, 2011 by TaoRich

[John Cuthber]

When the ball is in free fall, ie. clear of the ground, it will follow a parabola. That's fairly close to a sine wave, but not quite.

When it hits the ground things are much more complicated.

Also I think the "corners" on the curve you have shown represent an infinite rate of change of acceleration and I don't think that's realistic.

[Casey]

It would look like a "parabolic wave". It would be periodic with respect to time due to the lack of damping in the system. If you're looking for a "nice" equation, you're out of luck. However, I found a link about chaos theory and bouncing balls. It may not solve your problem, but it looks very cool: http://chaos.phy.ohiou.edu/~thomas/chaos/bouncing_ball.html.

 

At the end points (when it hits the ground), it would be an in spontaneous change in motion just like the graph of the absolute value function. The magnitude of velocity would be the same just before and just after impact.

 

In a computer algorithm, you could iterate the regular equation of motion over and over. In the algorithm, start on the ground going up (from the origin). The max of the parabola is the initial height from which the ball was dropped, so this is where the actual physical process starts. When the ball hits the ground, there will be a zero in the graph. Now, the second parabola will begin. It will have the same height and "spread" as the original. Start the new parabola from the zero that the original ended at and calculate its path to the next zero. Repeat ad infinitum.

 

I hope this helps. If you've ever studied NKS, you'll be familiar with the limitations of mathematical models and the superiority of computer algorithms (that was a hint. Check out NKS!). Based on absolutely nothing except intuition, I think the path could be approximated by the absolute value of an elliptic function, which is a more general version of the trig functions we're all used to. I imagine this would be very difficult, though.

[Xittenn]

Maybe you might like to look at something like the Lennard-Jones Potential and expand on it to fit your model. It can very well represent a perfect bounce where all energy is returned. I can't elaborate on this, I haven't focused enough time on the subject to do so. Is just a thought ...

[TaoRich]

Thanks for the input.

 

I realised after I went to bed last night that I didn't make it clear that i was referring to a ball bouncing vertically up and down on exactly the same impact spot, and not a ball bouncing with transverse motion as well as vertical. Mentioning it now for clarity sake, but I think that most of you got that first time around.

 

When the ball is in free fall, ie. clear of the ground, it will follow a parabola. That's fairly close to a sine wave, but not quite.

Do you mean that the movement path will follow a parabola or that the rate of fall/climb will be parabolic ?

If the gravity is a constant force, apart from the turnaround bounce point, surely the behaviour is the same as a SHM spring oscillator ?

 

When it hits the ground things are much more complicated.

Yep. We'd have the ball squishing and deforming, absorbing kinetic energy, and then "expelling" that energy again. In golf ball design alone, that alone is probably a multi-million dollar annual research exercise.

 

Also I think the "corners" on the curve you have shown represent an infinite rate of change of acceleration and I don't think that's realistic.

Oh, I Gimp-hacked a Wikipedia Sine Wave just to illustrate a point.

 

At the end points (when it hits the ground), it would be an in spontaneous change in motion just like the graph of the absolute value function. The magnitude of velocity would be the same just before and just after impact.

Which is (to my mind) the same as reflecting the sine wave ... open to opinion/correction.

 

Maybe you might like to look at something like the Lennard-Jones Potential and expand on it to fit your model.

Luckily, I don't actually need this for my "thoery paper".

 

At this stage, I was looking for a "illustrative example" to simplify my initial description of my expanding/collapsing fundamental space cycle.

 

My modelling is truly sinusoidal SHM, but the part which drops below the line requires me to bring in the concept of an anti-space too early in my discussion.

 

I need to make an initial gross simplification of the expansion/contraction cycle to discuss the "mechanics of the conservation of space", before i drag the reader into a fuller and more complete description of the true cycle (which will require some leap of the imagination).

 

I'm not sure how much of this makes sense (it's probably off the wall for most people) but the comments and opinions are really helping me.

 

So again - much appreciated.

Rich

Edited February 21, 2011 by TaoRich

[swansont]

You could look at this as if it were a solution with both positive and negative values of y and then rectify the answer. This models the impact as being instantaneous. In that case, the force on the ball is always F = -mg (g here is a scalar), which is Hooke's law and means the solution will be identical to that of a spring, i.e. the sine wave, and then you take the absolute value of it, as you suspected.

[TaoRich]

You could look at this as if it were a solution with both positive and negative values of y and then rectify the answer. This models the impact as being instantaneous. In that case, the force on the ball is always F = -mg (g here is a scalar), which is Hooke's law and means the solution will be identical to that of a spring, i.e. the sine wave, and then you take the absolute value of it, as you suspected.

 

Lovely .. thanks ... helpful

 

I'll bend the word and concept of "absolute" to fit into my explanation.

 

Above 'the line', I have "a space designated as alpha" which "interacts with a second space designated as beta".

 

It's this initial interaction that I need to explain first (to my audience).

I need to get the understanding of the "same world" interrelationship clear.

 

Hence my desire to simplify things at the outset for the first pass across the model.

 

Once this is clear, I'll move on to the next step:

 

Below 'the line', I have complementary relationships "(anti)space anti-alpha" "interacts with (anti)space anti-beta".

 

That's too much to get across/grasp in one pass.

 

 

It hurt my head for a few years trying to come to grips with that myself !

 

 

 

 

[swansont]

Now that I'm fully awake and caffeinated, I realize my previous statement was wrong. Hooke's law is F=-kx, so you will get a non-sinusoidal answer for gravity.

[TaoRich]

Hooke's law is F=-kx, so you will get a non-sinusoidal answer for gravity.

I've managed to solve my problem. I was thinking too loosely and grasping for an analogous illustrative example, when I should have rather tried to think more tightly and discuss what I have at hand, instead of making flawed parallels.

 

For completion sake however, I wanted to get a proper understanding of my own bouncing ball question, and a decent answer.

 

So here goes:

 

http://en.wikipedia....wiki/Hookes_law

 

Hooke's law: Ut tensio, sic vis, meaning, "As the extension, so the force".

The force exerted by the spring in SHM increases with the magnitude of the stretch or displacement.

Even though gravity is a constant force pulling down on the sprung weight, the spring force upwards varies.

 

So that is not the same as the (constant) force of gravity pulling on a bouncing ball.

 

Looking now at a pendulum:

 

http://en.wikipedia.org/wiki/Pendulum

 

Mathematically, for small swings the pendulum approximates a harmonic oscillator, and its motion approximates to simple harmonic motion

Again, we don't have true SHM.

 

Thanks for the comments.

All helpful as usual.

Edited February 21, 2011 by TaoRich