this is a copy of post 282 but with diagrams!!!!!!!
For Yandross, this is a situation in which each floor is blown sequentially.
If d= distance from the level of initial collapse
it will take 0.78 seconds for the upper section to reach the next floor, 3 meters below initial collapse
and if the next floor is blown at t=0.39 seconds after the initial collapse begins(one half the time for the upper section to reach the next floor)
d=0.5(9.8)t
2 for upper section
d= 3 + 0.5(9.8)(t-0.39)
2 for the next floor
The upper section mass will meet the floor and columns of the next floor when both of these are equal.
0.5(9.8)t
2=3 + 0.5(9.8)(t-0.39)
2
4.9t
2=3 + 4.9(t-0.39)
23=4.9t
2 - 4.9(t-0.39)
23/4.9 = t
2 - (t-0.39)
20.612 = t
2 - (t-0.39)
20.612 = t
2 - t
2 + 0.78t - 0.152
0.612 = 0.78t - 0.152
t=0.979
Self check:
At 0.979 s the upper section has fallen
d= 0.5(9.8)(0.979)
2d= 4.70meters
At t=0.979 the second level has been falling for (0.979-0.39) = 0.589 s
d = 3 + 0.5(9.8)(0.589)
2d = 4.70 meters
The upper section has collided with the next level's floor and column mass
The upper section is moving at v=at
v=9.8(0.979) =9.69 m/s
The mass of the second floor is moving at
v=9.8(0.589) = 5.77 m/s
_____________________________________
Now conservation of momentum
Assume that the upper section is 10 times the mass of one floor.
The upper section's momentum at the time it meets the falling mass of the next floor is;
p
u=10m(9.69)
whereas the momentum of the next floor's falling mass is
p
n=m(5.77)
If they collide inelastically then by conservation of momentum we have
p
t=p
u + p
n
p
t=10m(9.69) + m(5.77)
Now we know that p
t=11mv where v is the velocity of the now combined masses so;
11mv=10m(9.69) + m(5.77)
11mv=m(96.9 + 5.77)
v=9.33 m/s
The total mass is now slower, by 3.7%, than the original upper section would have been in an unimpeded free fall.
You can plug in any numbers you want about when the next floor is blown but the velocity of the upper block will always be greater than anything that starts moving at a later time and therefore will always catch the slower mass. In fact the worst case senario for catching that mass that starts moving after the upper block does, is the case where the next floor gets blown just as the upper block gets to it. In that case, after a drop of 3 meters the upper block would be doing 7.64 m/s and the next floor's mass is not yet moving(its has been blown from its connection to the lower structure just as the upper block reachs it) so its velocity and momentum is zero.
Now the combined mass of 11m will have a new momentum (as per conservation of momentum)of
11mv=10m(7.64) + m(0.0)
giving us a velocity of the combined mass of
v=(10/11)7.64
v= 6.95 m/s
a 9% drop in velocity from what the upper section would be doing if it had been in unimpeded free fall.