Started with 15 stories, and the impact/failure of the floor below adds to the falling mass:
p = momentum = m x v
m1 = mass of the top 15 stories
m2 = mass of the top 16 stories = aprox. (16/15) x m1
v1 = velocity before the additional mass is added = 8.63 m/s
v2 = velocity after the mass is added (unknown)
Momentum is conserved, so:
p = m1 x v1 = m2 x v2 = (31/30) x m1 x v2
Solving for v2:
V
2=V
1*(15/16)
V
2=8.63*(15/16)
V2~ 8.09 m/s
We want to find DeltaT for an accurate calculation of the instantaneous acceleration caused by impact with the floor so:
d = 1/2 (Vi+Vf) x t
d=.46 meters Vi=8.63 m/s Vf=8.09m/s t=?
0.46=1/2(8.63-8.09)t
0.46=1/2(0.54)t
0.46=0.27t
t= 0.46/0.27
t= 1.7 seconds
A=dV/dT
dV=0.54 m/s dT=1.7 seconds
A=0.54/1.7
A= 0.31 m/s2This means the floor fails by an overwhelming margin since the upward resistance only reduces the acceleration by 0.31 m/s/s when the force hitting the floor is 7.99 times that when the mass is at rest.
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Since the floor failed we have to restart the equation to account for the initial velocity (8.09 m/s) falling another floor height (3.8 meters) and find out what the maximum possible force exerted on the next floor will be
16 stories worth of mass is the new starting point for the North tower falling on the floor below
d=distance, g=acceleration of gravity, t=time, v=velocity
d = 0.5 g x t²
Solve for time when g=9.81 m/s2, d= 3.8 meters:
3.8=0.5(9.81)*t
23.8=4.905*t
23.8/4.905=t
20.775=t
2(SQRT)0.775=t
0.88 Seconds = t (again)
Solve for v:
V
F=V
I+gt
Since initial velocity is now
8.09 m/s due to the dampening effect of the first impact:
V
F=8.09+(9.81*0.88)
V
F=
18.78 m/s
- So the VF value was 8.63 m/s by the moment of impact with the first floor.
- You lost 0.54 m/s with the resistance of the floor slab before it gave out, giving a new VI for the next impact of 8.09 m/s
- Then for the next floor impact your VF increases to 18.78 m/s
The math shows a net gain in velocity. So let's recalculate the instantaneous acceleration once more. Assuming a floor has the capacity to stop the mass
completely. (DeltaV=-18.78 m/s). Depth of the floor is assumed to be 18 inches or ~.46 meters. Find time:
d = 1/2 (Vi+Vf) x t
D= .46 meters Vi=18.78 m/s Vf=0 m/s t=?
.46=1/2(18.78+0)*t
.46= 9.39t
t= .46/9.39
t= 0.049 seconds
Acceleration:
A=deltaV/DeltaT
Since we count the direction of gravity as positive and the force required to stop the mass is in the negative direction dV=-18.78 m/s
dT=.049 seconds
A=-18.78/0.049
A= -383.265 m/s2In other words the dynamic load at the 2nd floor impact is
~39.06G
But again, the floor's capacity as estimated by NIST is about 6x the static load. And our calculated dynamic load is
6.5x that. So it's apparent that the floor again doesn't have nearly the capacity to bring the mass to a
stop. Meaning it fails before an
~39g amplification is met. Let's find something closer:
p = momentum = m x v
m1 = mass of the top 16 stories
m2 = mass of the top 17 stories = aprox. (17/16) x m1
v1 = velocity before the additional mass is added = 18.78 m/s
v2 = velocity after the mass is added (unknown)
Momentum is conserved, so:
p = m1 x v1 = m2 x v2 = (31/30) x m1 x v2
Solving for v2:
V
2=V
1*(16/17)
V
2=18.78*(16/17)
V2~ 17.675 m/s
We want to find DeltaT for an accurate calculation of the instantaneous acceleration caused by impact with the floor so again:
d = 1/2 (Vi+Vf) x t
d=.46 meters Vi=18.78 m/s Vf=17.675 m/s t=?
0.46=1/2(18.78-17.675)t
0.46=1/2(1.105)t
0.46=0.55t
t= 0.46/0.55
t= 0.836 seconds
A=dV/dT
dV=1.105 m/s dT=0.836 seconds
A=1.105/0.836
A= 1.32 m/s2
And as before, since the direction of gravity is considered positive, the upwards direction will be considered negative here so the value for acceleration relative to the direction of gravity in this final value should read:
A= -1.32 m/s2