Copyright Kristin C Sabo 1997 The Falcon's Cannon ~~~ ~~~~~~~~ ~~~~~~ Time for a little more television physics, courtesy of moi, once-upon-a-time known as The Axe-Physicist of Louisiana. This time we're gonna play with TNot Falcon, and the Falcon's amazing "falcon-shaped" (no, J-Squared, I had no idea it was falcon-shaped! <> <>) ~canon~. A-hem, Jim and Artie did us a favour in act two. Artemus figures out that the 3 (yes, *three* J-J) towns the Falcon has already leveled - Salidas, Rome Plateau, and Tonga Flats, lie roughly on a curve with Sinful, CO in the center. He even shows Jim - and us - on a map, identifying the three targets as being in the SW, S, and SE portion of the state, with Sinful a bit north, above the middle town. So I've got my handy-dandy map of Colorado here. The state is nearly square, with a length of 375 km and a width of 275 km. Denver is 150 km in from the east and 100 km in from the north. That puts the mythical Sinful, CO between 75 km and 125 km somwhere south of Denver. Of course, Sinful is a name made up for the show, but funny enough, as I look on the map, in almost the perfect location, there is a *real* city called "Canon City". Heh. I'll take it at 100 km (60 miles) S of Denver. So how much force does Falcon's canon have to exert on a projectile to launch it from Canon City to Denver, if he's going to uphold his threat to our heroes? We aren't even going to look at what type of projectile or what type of propulsion is used for now. Time to do a bit of science! <> We know that the angle at which the projectile was to be launched was 72 degrees to the horizontal [Falcon told us], and it needs to travel 100 km or 100,000 meters. Let the initial speed of the projectile, at an angle of 72 degrees = V. Projectile physics has been known for centuries, and simply requires some trigonometry. Trig says that along the ground component, the initial speed of the projectile is V(cos 72 deg) = .3V , and straight upwards is V (sin 72 deg) = .95V . There is this thing, called The Range Equation, that folks usually just look up to determine how fast something has to be launched to reach a certain distance, if the angle at which it is launched is known. I of course had to rederrive it, but it ain't hard... just algebra (which is always what I screw up.) RANGE = V(cos 72 deg) x [-4V(sin 72 deg) where g = ----------- the acceleration g ] of gravity on Earth = -9.8m/s^2 So RANGE = 100,000 m = [.3V x .4V]s^2/m - the units work out, thank goodness. and V = sqrt[100,000] --------- m/s = 2635 meters per second = 2.6 km per sec (.3x.4) (Well, at least this is less than the earth's escape velocity, which is 11.2 km/s) In units more familiar, the velocity at which the Falcon's projectile is launched, V = 5760 miles per hour... *smokin'!* This means that if one of our heroes got loose just in time to warn Denver, they'd have 2.1 minutes from launch to send a telegraph and all the citizens to get out of town. I don't think so.... Okay, so it's big. Muzzel velocities of handguns are on the order of hundreds of miles per hour, a factor of 10 lower. Doesn't seem ridiculous to me, but then I have no idea of the average canon muzzel velocity. Anyone know? What I find inpossible is the maximum altitude achieved by the shell. Using kinematics again, we get the max height achieved = 177,725 m = 10 miles, roughly. The equations I used consider the acceleration of gravity to be a constant, and at these astounding heights, that no longer holds. Falcon is practically putting his shell into orbit! It may be interesting thing is what happens when you accelerate the projectile down the canon barrel to launch it with that speed. The length of the barrel looks to be approx. 2 meters on screen. The projectile starts at rest and reaches 2635 m/s at the end of the barrel. Assuming uniform acceleration (ba-ad Kris), the time it takes to traverse the barrel is roughly .0007 seconds and the average acceleration along the barrel looks like 3.8 x 10^6 m/s^2, or 384000 "G"s. You've heard that before. They always talk about how much G-force someone pulls while flying a plane or spacecraft, or even in a car accident, where "G"s can go up into the 100s. Humans lose consciousness typically around 7Gs (prolonged exposure) without conditioning. I don't know that plain iron is going to hold together under the sort of impulse we're talking about here, either. You probably don't want to play human canonball with Falcon's gun, as Loveless did in that awful episode with Jeremeelzebub. I'd do more, but we haven't a clue what kind of projectile the Falcon gun fires. If we believe Jim and Artie, and I guess we should, it's actually some form of shell. But what size??? Could weight anything from a few pounds to a few thousand pounds. And what is he using for combustion? Sigh. Such unanswered questions.... -Trail Mix (<> who finds physics to be much better than a cold shower... <> zzzzzzzzzzz....zzzz....) -- ............................................... ksabo@physics.usc.edu .....