THE ORBITER “RAN INTO” THE FOAM
“How could a lightweight piece of foam travel so fast and hit
the wing at 545 miles per hour?”
Just prior to separating from the External Tank, the foam was
traveling with the Shuttle stack at about 1,568 mph (2,300
feet per second). Visual evidence shows that the foam de-
bris impacted the wing approximately 0.161 seconds after
separating from the External Tank. In that time, the velocity
of the foam debris slowed from 1,568 mph to about 1,022
mph (1,500 feet per second). Therefore, the Orbiter hit the
foam with a relative velocity of about 545 mph (800 feet per
second). In essence, the foam debris slowed down and the
Orbiter did not, so the Orbiter ran into the foam. The foam
slowed down rapidly because such low-density objects have
low ballistic coefficients, which means their speed rapidly
decreases when they lose their means of propulsion.
The force due to drag on an object is described by the following equation (http://en.wikipedia.org/wiki/Drag_equation):
F = (1/2)p(v^2)(C_d)A
where p is the density of the fluid the object is moving in, v is the velocity of the object relative
to the fluid, C_d is the coefficient of drag and A the cross-sectional area. C_d is around 1 for most geometries (http://en.wikipedia.org/wiki/Drag_coefficient).
In the CAID report, the dimensions of the foam is estimated to be 19x11.5x11.5 inches, which is roughly 0.5x0.3x0.15 meters. Assuming the foam fell with the smallest surface pointing down (a bad assumption), we can estimate A to be 0.045 m^2.
According to Wolfram Alpha, the density of air at 20km, the altitude at which the foam broke off, is 0.089 kg/m^3. The initial velocity of the foam was about 1568 mph which is around 700 m/s.
Plugging all of this in to the above equation gives:
F ~= 981 kg*m/s^2
The mass of the foam was estimated at 1.6 lbs, or around 0.7 kg, and using F=ma we can calculate the acceleration to be about 1400 m/s^2. The foam traveled for 0.16 seconds before striking the shuttle, and assuming that the above force was constant during that time (thus overestimating the change in velocity), that would result in a delta_v for the foam of 224 m/s, which is about 500 mph.
This is all assuming of course I didn't make some trivial mistake with the math, which I probably did.
It would be great if someone with more knowledge than me could chime in about drag near the speed of sound (or multiples of the speed of sound).
I'm also extremely curious to know how much the Shuttle stack accelerated during those 0.161 seconds the foam wasn't attached to it...
Most of the mass in a rocket is fuel. As the fuel is burned off, the same amount of thrust gets divided over a progressively lower remaining mass.
The Shuttle stack has a mass of 2.03 million kg at launch, and delivers 30.45 MN of thrust. That's 15.0 m/s² of acceleration, or 1.53 g's. (Because it's thrusting vertically, though, it only accelerates upward at 0.53 g's.) This is much lower than the maximum acceleration attained later in flight.