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Heiwa is just not getting this. I can't tell of its stubbornness, gamesmanship or just lack of aptitude. But this is for anyone else who is trying to pick apart these concepts. This post & the next one will allow you to really understand well, at a fundamental level, what is going on.
And also where Heiwa is blowing his "analysis".
Heiwa is NOT getting the concept of "compacted". Compacted is approximately equivalent to "broken down, lying flat, still in the box". Dropping one piece, then another, then another is NOT equivalent to compacted.
When you drop a compacted bookcase, you do NOT get "bang, bang, bang". You get one, LOUD "bang". If the components of the bookcase are tightly pressed together, the forces generated can be substantially greater than those generated by the impact of the assembled bookcase.
First, we have to compare the impact of any rigid object versus any flexible object colliding with a rigid surface, like the ground.
We will use a Force versus Time graph to show the evolution of the (equal & opposite) forces that are generated between the colliding bodies.
One thing is absolutely crucial to understand about these curves. In the case of a collision with the ground (as we will discuss below), the total area under each curve is determined BEFORE THE COLLISION ever takes place. Since the final velocity is zero, the final momentum (m*v) is zero, and the change in momentum is completely determined by the initial mass & velocity of the components. For two objects of the same mass, moving at the same speed before the collision and at rest after the collision, the momentum change is exactly the same. It does not matter whether the time interval for this process was long or short.
The analysis is only slightly more complicated for a collision between two moving objects. We won't go into that.
Looking at the curves below, it is easy to see that the main difference between the two collisions is the time duration of the collision. Any flexible object, like a soft rubber ball, draws out the collision in time. Any rigid object, like a baseball, produces a very short duration collision.
Two such collisions are shown in the Force vs. Time Fig. 1 below. Note that the units on both axes are arbitrary and for comparison only.
[qimg]http://www.internationalskeptics.com/forums/picture.php?albumid=176&pictureid=1377[/qimg]
Figure 1. Comparison of the collision of a rigid object versus a flexible object.
Notice that, since the area under the two curves is exactly the same, the simple act of drawing the collision out in time reduces both the peak force and the average force generated by the collision.
Let's apply this insight to the fall of a bookcase. As before, the bookcase has 4 pieces: the shell (the top, bottom, sides & back) and 3 shelves with books. For simplicity, I've made the mass of each piece be the same.
Using the curves in Fig 1 to describe the bookcase collision with the ground, the flexing of the assembled bookcase results in a "flexible collision", drawn out in time.
The other collisions (the books & shelves hitting the ground, and the packaged bookcase hitting the ground) are considered rigid, short duration collisions.
There are 3 cases to consider:
Figure 2 shows the force versus time graph that would be obtained for each of the conditions described below.
[qimg]http://www.internationalskeptics.com/forums/picture.php?albumid=176&pictureid=1378[/qimg]
Figure 2. Bookcase collision
Case A (blue line): The bookcase is broken down, flat and disassembled, the books stacked neatly on top, and the whole bundle is wrapped with packing tape into a nice, tight bundle.
Case B (green line): The bookshelf is assembled with the books on it. When the bookcase hits the ground, the whole frame and shelves all flex like crazy, but all components manage to withstand the impact without breaking.
Case C (red line): The bookcase is assembled with books on it. When the bookcase hits the ground, each of the shelf support brackets break when the load has grown to 0.05 (force units). This means that the total force generated by each shelf will rise to that level during the initial impact, just like in Case B. But suddenly each of the shelves will break free and the force component from the shelves will drop to zero.
The important thing to realize is that, once again, the total area under each of these curves is the same. That area is determined before the collision occurs by the initial mass & velocity of the parts. It has nothing whatsoever to do with the stiffness or flexibility of the components, or for that matter, whether they are joined into one piece or in a million pieces.
The time duration of the impact is determined by the flexibility of the various components, and by the spatial distribution of the parts. As you can see, if parts are separated in space, then their collisions are also separated in time. As shown in Case C, fractured components can result in collisions that are separated in time.
If these principles are starting to make sense, then you're approaching a deep understanding of how collisions really work.
I'll save that one for the next (& last) post in this series. It'll be a fun one.
Tom
