Rideman said:
"Objects falling near the earth's surface tend to accelerate at a rate of about 32
ft/sec/sec. With this bit of information, you can take indefinite integrals and get:
a = 32 ft/sec/sec.
v = 32t + v0 feet/second
d = 16t^2 + v0t + d0 feet
If we assume that d0=0 (we're falling the whole distance) and v0=0 (we're falling from a
dead stop) we can reduce the equations as follows:
a = 32 ft/sec/sec
v = 32t ft/sec
d = 16t^2 feet
If we know the distance (310 feet, for instance) we can solve for time:
d = 16t^2 feet
d/16 = t^2
sqr(d)/4 = t
And we can substitute that expression for t in the velocity equation:
v = 32t feet/second
v = 32 * (sqr(d)/4) feet/second
We can do one more conversion because 1 mph is equal to 5280 feet per 3600 seconds:
v = (3600/5280) * 32 * (sqr(d)/4) miles/hour
If you reduce all the fractions you end up with:
V [mph] = 60/11 * sqr (d [feet])
For a 310 foot drop, this works out to 96.037 mph.
It is reasonable to expect that the formula will give a higher number than reality,
because the formula represents an ideal case, where there are no resistive forces
(friction, wind resistance, etc.). But it will get you into the ballpark for most rides. It
also fails to take into consideration the initial velocity which, on Millennium Force, is 22
ft/sec. I didn't manage to work that one out because when v0 is non-zero, the
equations get complicated:
a = 32 ft/sec/sec
v = 32t + v0 ft/sec
d = 16t^2 + v0t feet
This means that the time required is now a quadratic polynomial and I always hated
solving those. I got as far as...
310 = 16t^2 + 22t
155 = 8t^2 + 11t
155 = t(8t+11)
If anybody wants to solve that one for t, then you can plug it into...
v = 32t + 22 ft/sec
then multiply by 3600/5280 to get the ideal maximum speed that Millennium Force can
possibly attain in an ideal world."
--Dave Althoff, Jr.
Bill Said:
"It does the hill in 22s, right?
300 vertical feet at 45 degrees = 486ft = 148.15m
in 22s = 6.73m/s
(ignore mass, as it cancels)
Ep0 = energy = gh = 9.8*91.46
Ek0 = .5v^2 = (1/2) * 6.73^2 = 22.67
Ebottom = 1/2v^2
.5v^2 = 918.978
v^2 = 1837.96
v = 42.87m/s = 95.897mph"
Here's the link to the thread. BEWARE...it is summer break and there are several mathematical equations in this post. Reading this thread could cause your brain to hurt...
http://www.guidetothepoint.com/thepoint/cpplace/thread.asp?ForumID=2&TopicID=1783