[ Note: this text is altered from its posted form by updating my email address to sheol.org, and the hyperion site URLs to point to www.midwinter.com. Also: the comment from Gharlane is a bit out-of-context; he and I weren't disagreeing on anything substantial, and his comment was directed to somebody else. I was just using his comment as a jumping-off-point to put numbers to this situation. ] From: throopw@sheol.org (Wayne Throop) Subject: Re: the fall in the Fall Date: 17 Nov 1995 18:37:12 GMT Message-ID: <48ikoo$u5@aurns1.aur.alcatel.com> : From: gharlane@ccshp1.ccs.csus.edu (Gharlane of Eddore) : You're being silly; you speak (pardon me, type!) as though you feel : Sheridan would be accelerating due to some kind of gravity as he : "falls" from the core shuttle to the interior surface.... : THIS IS NOT THE CASE. True. He is not accelerating due to some kind of gravity. But he IS accelerating due to some kind of wind. As he moves out radially, the wind picks up tangentially, accelerating 'im. This in turn (in the air-stationary reference coordinate system) produces centrifugal and corriolis effects. Or, (in an inertial coordinate system) being accelerated at a tangent to a small radius means he's also accelerated towards the larger-radius rim. If the wind picks up to 120mph, he'd be accelerated at 1g, which would (in the inertial frame) quickly get his windspeed back under 120mph. Since the air at the floor of the garden is still with respect to the ground, he just can't splat in at more than 120mph. Well, unless he's *trying* to make things worse (see below in "auxiliary comments"). Now, I'll grant you that what Ivanova said is pretty much incompatible with what JMS has said as Revealed Truth about the garden. But it isn't as bad as all that. Let's go back and review. First, let's gather information and attribute sources. First, the Revealed Truth about the garden size, Handed Down by JMS Himself: By virtue of being patterned physically after the work of such scientists as Gerard K. O'Neill, the absolute center of the elongated station (which revolves to provide gravity) is a sort of hollow-world look, with fields and hydroponic gardens along the 360-degree circular section (which is about a half-mile, or a mile across)...and as you get closer to the absolute center, where a transport tube cuts from one end of the station to the other, naturally you get less and less gravity until you can literally hang suspended. This area is known as the Garden. --- http://www.midwinter.com/lurk/universe/station-1.html Now, I wasn't able to find a definite source of Revealed Truth about the rotation rate of the garden, but it is fairly decisively portrayed as rotating with a total period of between 40 and 60 seconds. In any event, periods below 30 seconds are Right Out, at least, according to the location of the garden in the station map at http://www.midwinter.com/lurk/universe/station-map-1.html coupled with the exterior rotation shots. So let's say 30 to 60 seconds to be generous. Now, what did Ivanova say? "He's more or less weightless," Ivanova explains. "But the ground is rotating at sixty miles an hour. If we can't catch him, he'll be killed by the impact." --- http://www.midwinter.com/lurk/synops/044.html Not in the synopsis, but she also says to ops on her link "You don't HAVE two minutes! You have thirty seconds!" So. Let's assume the longest plausible period and the smallest Revealed Radius for the garden. We get r=400 meters, p=60 seconds, a=0.45 g, v=41.9 meters/sec (94mph) So, right off the bat, we know that Ivanova cannot possibly be right about the groundspeed. Revealed Truth yields a groundspeed of 94mph at a minimum. Using the numbers at the other extreme, we have r=800 meters, p=30 seconds, a=3.5 g, v=168 meters/sec (376mph) We can, of course, rule this one out, since the largest Revealed Acceleration in the whole station is 1.2g. But using the more plausible p=60, we have 4=800 meters, p=60 seconds, a=0.9g, v=84 meters/sec (188mph) Now, if we presume air drag proportional to v^2, and a terminal velocity in garden air of 120mph, and simulate the r=400, p=60 case, we get an airspeed at impact with the rim of 22 meters/sec, or about 49mph, and a fall time of 70 seconds. If we simulate the r=800, p=60 case, we get an airspeed at impact of 37 meters/sec, or about 82 mph, and a fall time of 88 seconds. Both of these are compatible with the fall time between 30 and 120 seconds Ivanova's comments require, both are compatible with the Revealed Range of Radius for the garden, and they bracket a case with terminal airspeed of 60mph. [-- 54 m/s == 1g air drag --] [---- numbers for 60 sec rotation ----] [---- splatpoint numbers ----] rim accel radius fall time v(rim) v(air) v(tangent) v(radial) --------- ------ --------- ------ ------ ---------- --------- .45g 400 m 70 sec 42 m/s 22 m/s 16 m/s 14 m/s 94 mph 49 mph 35 mph 31 mph .60g 536 m 78 sec 56 m/s 27 m/s 19 m/s 20 m/s 125 mph 60 mph 43 mph 45 mph .80g 715 m 85 sec 75 m/s 34 m/s 21 m/s 26 m/s 168 mph 76 mph 47 mph 58 mph .90g 800 m 88 sec 84 m/s 37 m/s 22 m/s 29 m/s 188 mph 82 mph 49 mph 65 mph If we go from the diagram at http://www.midwinter.com/lurk/gif/map.gif and count pixels from the radius to the garden floor, and then assume that the outermost rim of the station under the garden has 1.2g, the garden floor itself has .8g or so, which puts us splat, er, "right" in the middle of the above table. So, while I agree that Ivanova's statement that "But the ground is rotating at sixty miles an hour" is at best misleading, and definitely technically incorrect as it stands, she immediately follows up with "If we can't catch him, he'll be killed by the impact", which may well indicate that she's talking about either the expected groundspeed or airspeed accounting for air drag. And I STILL maintain that such an interpretation is entirely consistent with what we know of the physics which would exist on an actual station. The bottom line: JMS didn't make that much of a physics gaffe here, if any. For one thing, it is within established possibility that the rimspeed is 95 mph or so (which isn't all that far off from 60mph), and it is entirely reasonable for Ivanova to be taking air drag into account, in which case 60 mph is a Pretty Darn Good Number. Some auxiliary points. I've been told that a skydiver can slow to 90mph or speed to 200mph using the same (vanilla) jumpsuit. This has some implications about what Sheridan might do to save himself. At the 1g drag at 90mph and a 0.6g garden, the splatspeed is still v(air)=54mph or so. It doesn't help that much; the more efficiently one reduces one's tangential velocity during the fall, the higher the radial velocity at splatpoint. Sigh. However, for the first 10 seconds or so, you only need to "swim" in a tenth-g relative to the air to make progress back to the axis. And once you get back to the axis, you can rest... even grab onto the shuttle tramway. Therefore, it seems to me that Sheridan's best strategy is, quick as he can, get out of his jacket at least, and maybe his pants, and FLAP THEM LIKE MAD to try to "swim" up. He actually seems to have a chance at it, if he can get his jacket off in (say) less than 5 seconds. (Point of diminishing returns if he tries to strip off his pants, unless he's very practiced at it.) Even if he has no chance to make centerward progress, he may very well push his fall time out beyond the two minutes the rescue team quoted Ivanova. Specifically, the 70-second-or-so fall times are assuming 5 m/s velocity jumping out of the shuttle. If he can merely slow that down to an effective rate of 1 m/s, then even if he drops out of the "sweet spot" in the center of the cylinder and can't maintain his position, he's still lengthened his fall time to 135 seconds or more in the 0.6g case. If he then tries to avoid air drag as much as possible (using the 200mph terminal max airspeed quoted above) he can prolong his fall to 160 seconds or more in the 0.6g case. His splatspeed goes up to 76 mph (from 60 mph), but that's what's being traded off for the longer hang time. In the 0.8g case, the hang time goes all the way up above 170 seconds, close to three minutes. So, all in all, if you are ever on B5, and (for some reason) jump from the shuttle, your best strategy seems to be to try like hell to "swim" back up to the zeropoint. If you fail, and notice the breeze rising, cease flailing and adopt the classic skydiver's I-wanna-go-fast pose (which, paradoxically, KEEPS you from going fast in this case, when looked at inertially), with limbs clenched in close, and head (or feet) into the wind. That exploits your chance to reach safety (the zeropoint) yourself, and extends your hangtime if you fail and fall into fast-moving air. Well.... arguably anyway. Depends on whether 50+ mph seems survivable. In the 0.6g case, if you can't reach the zeropoint, it's right on the borderline. You can get your splatspeed down to (maybe) 50mph or so, or your hang time up above two minutes. But not both. Another strategy MIGHT be to try to exploit lift, and "body surf" to extend hang time. The simple model of drag I have can't quite handle that, but I'd be delighted if somebody else would try to model it... For another subject entirely, in counting pixels on the station diagram at http://www.midwinter.com/lurk/gif/map.gif I conclude that C&C has maybe 0.3g, based on the same assumptions that gives the garden 0.8g. The map.gif isn't guaranteed to-scale, but it all hangs together nicely, yielding plausible numbers. -- Wayne Throop throopw%sheol.uucp@dg-rtp.dg.com throop@aur.alcatel.com -- # simple radial-coordinate-system model of fall from # center of b5's open area above the zen garden to the rim # # these calculations are still quite crude, but seem to give reasonable # results. for example, as a simple check, setting air resistance to # zero, the vr remains constant all the way to the rim, and the vt is # within 1% of rimspeed (Of course, that's a degenerate case, but shhhhhh!) # parameters set tcl_precision 8 set pi [expr 2.0*asin(1.0)] set p 60.0; # rotation period of station in sec set g [expr 9.8*0.8]; # gravity at rim in m/s set d 0.0; # initial distance from center set vr 5.0; # initial radial velocity in m/sec (WRT still air) set vt 0.0; # initial tangential velocity in m/sec (WRT still air) set vterm 54.0; # terminal velocity in zen garden air in accel of 9.8m/s^2 # some derived numbers set vk [expr 9.8/($vterm*$vterm)] ;# air drag constant set r [expr ($p*$p*$g)/(4.0*$pi*$pi)] ;# station radius set v [expr (2.0*$pi*$r)/$p] ;# velocity of rim # startup display xd format "p=$p g=$g d=$d vr=$vr vt=$vt vk=$vk r=$r v=$v" set t 0.0 set dt 0.1 ;# time increment for simulation set out "" while {$d<$r} { # iterate until impact with rim set t [expr $t+$dt] set vti [expr $v*($d/$r)-$vt] ;# tangential velocity inertial set vair [expr sqrt($vr*$vr+$vt*$vt)] ;# total air-relative velocity set aw [expr $vk*$vair*$vair] ;# wind resistance set awr [expr $aw*($vr/$vair)] set awt [expr $aw*($vt/$vair)] set agr [if $d>0 {expr $vti*$vti/$d} {format 0.0}] ;# centrifugal effect set agt [expr ($v*(($d+$vr*$dt)/$r)-$v*($d/$r))/$dt] ;# corriolis effect set d [expr $d+$vr*$dt+.5*($agr-$awr)*$dt*$dt] set vr [expr $vr+($agr-$awr)*$dt] set vt [expr $vt+($agt-$awt)*$dt] append out [list t=$t d=$d vr=$vr vt=$vt vair=$vair aw=$aw \ ag=[expr sqrt($agr*$agr+$agt*$agt)]]\n } # result display xd format $out