(TFT) Math of Demon Towers --> Too expensive!

[Pasha and or Rick Smith <pnrsmith@istar.ca>]

Hello, 
	Here is my take at the demon tower.

	acceleration = 9.8 m / (s*s)

The velocity with constant acceleration is:

velocity at a given time
	 = starting velocity + acceleration * time

Assuming that the escape velocity is 11.2 km / s and
converting everything into meters we get:

11200 m / s = 0 m/s + 9.8 m / (s * s) * ? s

Solving for ? gives us:

(11200 m/ s) / 9.8 m/(s*s) = ? s  = 1142.9 s

(This is ignoring air resistance and changes in the
gravitational field between the top and bottom of 
the tower.)

Now how far would you fall in the 19 odd minutes
above?


The distance equation is:

distance (in a time) = starting distance + 
		( starting velocity * time ) +
		( 1/2 (acceleration) * time squared)

d( t ) = 0 + 0 + 1/2 * (9.8 m/(s*s) ) * 1142.9 s * 1142.9 s

d( t ) = 6,500,480 m

	Assuming the gates are 1,000 meters apart
I calculate it will pass thru the gates at least 6,501
times.

	The first time the gate fails, it will flicker 
for 1 minute before collapsing.  (In my campaign 
it lasts for an AVERAGE of 1 minute but might fail
after 5 seconds or might last a maximum of 1 minute
and 55 seconds (but that is just because I'm a mean
GM)).  However if a gate fails a second time while it
is flickering I rule that it fails instantly cutting the
person in half.  (I do it this way because it make 
going thru failing gates much more interesting 
and because I'm mean.)

	Now just before launching the demon into
orbit, they are passing thru 11 gates per second.  It
takes 5 seconds to cast the spell to stabilize the gate.
I believe a gate fails on a 17 or an 18.  This is a 4/216
chance or 0.0185... or just more than 1.85 %.

	Now assuming an average velocity of 
11 km / s just before orbit insertion our demon
will pass thru 55 gate transitions before the gate
can be repaired.  The probability of a second
gate failure given that one happens in the last
5 seconds or so is:

P(2nd gate failure) = 1 - ((1 - 0.0185 ) to the 55th power).
		   =  64.2 % (approximately)

	Of course this could happen on an earlier
5 second group of time, but many fewer gates would
be transited so the odds are much better that the
wizards could repair the gates. 

	Therefore, I wouldn't send any demon
I want to see alive again, into this device...

	Note that the demon would hit like a meteor
against the floor so any wizards standing around
to repair the gates are quite likely to be blown up.
That would increase the cost of such a device by a
significant fraction.

	A smaller version of this where a rock
only falls for 30 seconds or so could be safer and
would still make a powerful artillery piece for 
bombarding castle walls.  Such a device would
have the rock pass thru 5 or so gate and would
fire rocks at around 300 meters per second 
(neglecting air resistance).  Of course the wizard 
would have to get close enough to the castle wall
to cast the final targeting gate.

	One thing people have not considered
is the difficulty of hitting a 3m square target 
after falling 1000 meters.  How would we steer
our demon or rock while it is falling this 
distance?  If our missile touched the side of 
the tube it would rattle back and forth wiping 
out the tube, and if we make the fall a much
smaller distance (say 15 meters) then the
number of gate passages increases sharply
greatly increasing the cost.

	Rick
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