Reply #118: Maybe you need some help?

[Make7]

One of your objections to LARED's "model" is that he didn't use the center of the mass for potential energy. Which is a somewhat valid point, but you failed to use the center of the mass for kinetic energy in your "correction." Let's do some calculations using the center of the mass for both with LARED's equations:

   PE = KE + 1/3 PE
   PE = 3/2 KE
   mg(h/2) = 3/2 (1/2 mv²)
   v = (2/3 gh)^0.5

   v = 171 ft/sec

But here you used the full height for the rest of the calculation:

   t = 2 h / v
   t = 2 * 1368 / 171

   t = 16.0 seconds

(Please correct me if you arrived at your answer of 16 seconds differently.)

But here is what you should have done:

   t = 2 (h/2) / v
   t = 2 * (1368 / 2) / 171

   t = 8.0 seconds

So now we know how long the lower half of the building takes to collapse. (If 1/3 of the PE is used to overcome resistance.)

But who cares? That's not really very useful at all. Let's see if we can rectify that. (Isn't math fun!)

We know the time and distance for the first half of the collapse so we can easily calculate the rate of acceleration:

   a = 2 h / t²

Ohhhh...I almost forgot...it should be the height for the center of the mass:

   a = 2 (h/2) / t²

   a = 21.375 ft/sec²

Now I am just going to assume the rate of acceleration remains constant for the rest of the collapse. (After all, as LARED said, this is just a "very crude energy balance.")

Alrighty then - we know the velocity at the 8 second mark and we also know how much height is left of the collapsing building (not surprisingly it is h/2), let's find the final velocity:

   v_final = (v_initial² + 2 (h/2) a)^0.5

   v_final = 242 ft/sec

And now we can calculate the time it takes for the top half to fall to the ground:

   t = 2 (h/2) / (v_initial + v_final)

   t = 3.31 seconds

Here comes the interesting part - when I add the time for the bottom half and the top half together, I get 11.31 seconds. But the really interesting part is that I got 11.30 seconds with LARED's original equation when I used 1368 ft for the height and 32.16ft/s² for gravity. I must be doing something wrong.

Adding to the coincidences, that also matches the results I get with my simple spreadsheet comparing the rates of acceleration for different collapse times to gravity. Go figure. Please.

:) Make7