For a great introduction to rotating frames of reference, artificial gravity and space station issues, go to  "Artificial Gravity. Current Concerns and Design Considerations". The following discussion will be much easier after reading this interesting reference.
 

Imagine that you stand  on a slighly curved floor in a giant rotating space station (see Kubrick's "2001..." for reference). Let us assume that the station radius is 500 m and its rotational velocity provides "gravity" that is close to Earth's (9.81 m/s^2) at its outer diameter. The question is: if there were a fountain next to you, ejecting the water vertically (in the direction of the station's axis of rotaion), would the water's trajectory differ from what we can see on Earth?

The answer is  - obviously - yes, because of so-called Coriolis force, which would  act upon the moving water, curving its trajectory.

I wanted to find out exactly how the situation would look (as a part of research for a sci-fi movie script). The most elegant solution would be to derive the general equations of motion in rotational frames of reference, corresponding to Kepler's laws of orbital motion valid in our "normal" frame of reference. The gravity would be ignored in such considerations (I assume that a space station is small enough for its gravitational influence to be negligible), only the effects of relative motions would count. I started to think how to do it, but although I know how to derive Kepler's laws from the Newton's law of gravitation and his laws of motion, I could not figure out how to crack the rotating frame of reference "gravity" problem. I guess it is a simple case of having "inadequate mathematical tools" , and also not having enough time (this is not my main research after all). After a while I decided that it will be much easier to simulate the behavior of moving (thrown) objects in rotating space station using  a computer. That's how the simulation applet was created.

The  simulation is very easy to do after a little bit of thinking and realizing that (I ignore the air resistance and gravitational forces in my analysis):

1. Once an object is thrown in rotating space station, its trajectory in absolute frame of reference is essentially a straight line, whose direction depends on the initial velocity of the object. If the initial velocity is 0 (the object is simply dropped), its trajectory in absolute frame of reference is a straight line tangential to the circle passing through the object,with origin at the station center of rotation and perpendicular to the station's rotation axis. By "absolute frame of reference" I mean frame of reference tied to distant stars. I know that it is not really an absolute and inertial frame of reference, but it approximates one with enough accuracy. I simply want to avoid  an in-depth discussion of the Machian principle here (arguing what excatly is an absolute frame of reference and if it exists).
2. The object on the station seems to fall (as if under gravity) only because the floor, rotating in a circle, intercepts its straight-line path after the object is released or thrown. That is all that happens. The straight trajectory of the objects, combined with the circiular trajectory of the floor produce the entire gravity-like motion phenomenon for the observer tied to the rotating station. This is artificial gravity.

The entire simulation can be written using only these principles. There is actually no need to even introduce the concept of the Coriolis force. By the way, I think that Coriolis force is a little misleading concept. In my opinion, it should be probably called "apparent deflection of objects moving in rotating frames of reference" or something like that. In absolute frame of reference there is no force acting on a thrown object (in our space station). The object's trajectory  (as seen from the station) curves because of the circular motion of the space station. So there is only a relative motion problem here, no actual forces. The Coriolis force is an apparent force introduced to explain the curved paths of objects observed from rotating frames of reference. The word apparent should be always emphasized and always used whenever the Coriolis force is mentioned.

Just two observations about the bodies orbiting one another in rotating frame of refeence (assuming that an object is released or thrown from some point in the rotating frame of reference and then "orbits" it; I assume no walls or boundaries that could stop the object):

1. It seems to me that orbits are  always spiral-shaped (broadly speaking) and undbound  - since the released object moves in a potentially infinite straight line in absolute space and  its realease point (tied to the rotating frame of reference) always moves in closed circles in absolute space. My guess is that the orbits are third-degree curves (straight line combined with circle), but I'm not sure, they might as well be fourth degree (here is the "lack of mathematical tools" for you...).
2. The only case when the object's orbit is bound and not spiral-shaped accurs if the object is thrown in direction perpendicular to the station's radius and axis of rotation, with speed equal to the linear velocity of the release point (omega*radius_of_release_point) and in the direction opposite to the station's rotation. In this case the object will stop in absolute space and it will be circling through the station in the direction opposite to its rotation and "hovering'" always at the same distance from the floor - it will never fall. Its "orbit" with respect to the release point will then be a circle with radius equal to the radius of the release point, centered in the center of the station's rotation and  lying in the plane of the station's rotation.

Deriving these rotating frame of reference laws of orbital motion (equations for motion curves) might be an interesting term project in some astronomy or aerospace degree program. Unfortunately, I have no time to investigate this subject further (and  it is of no practical significance as far as I can tell).

Simulation

Anyway, the simulation simply assumes that the released (thrown) object moves in straight line in absolute space. Then the applet does a few coordinate system transformations (translation and rotation of the coord. system centered at the point on the floor directly below the release point) and voila. Everything is computed in 0.01 s time increments, offering some compromise between speed and accuracy. The computation ends when the object crosses the station floor (moves out of the circle with radius equal to the station's radius).

Here are the defaults for which the applet displays the trajectory when launched:

• Station radius: 500m. This  seems to be a reasonable value, large enough to reduce various physiological (balance and gravity gradient related) phenomena so that they are bearable to the inhabitants (see the reference at the beginning of this page for more details). 
• Release point: 1 m above the station's floor. This is done to make things more interesting. You can specify the release (throw) speed to be 0 and then you will see how an object falls on a rotating space station and is deflected by the apparent Coriolis force.
• Release (throw) speed: 7 m/s. This is a speed that can be expected in a typical fountain throwing the  water a couple of meters in the air.
• Release (throw) direction: 0 degrees, meaning shooting the water (or whatever) straight up, in the direction of the station's axis of rotation.

The station is assumed to rotate in the plane of the screen counterclockwise. You are standing on its "bottom", with the rotation axis above you (usually beyond the upper edge of the display window).

You can control the following parameters:

1. The velocity magnitude, from 0 (object falling down without any initial velocity from one meter) to 400 m/s (60 m/s faster than the speed of sound).
2. The velocity (throw) direction: from -90 (shooting to the left - or against the station's rotation - perpendicular to the station's radius) to 90 degrees (shooting horizontally to the right).
3. The station radius from 2 to 2000 m.

The station's rotational velocity is recomputed for each radius so that the "gravity" at the station's radius s equal to the Earth's gravity. The smaller the radius, the faster the station has to rotate in order to provide artificial gravity equal to the Earth's gravity, and the more pronounced the Coriolis apparent deflection.

The release point is marked with short horizontal line crossing the vertical red line, which represents the station's radius. The station's circumfeence is always drawn and at least partially visible at the bottom of the display screen as the red section of a circle. For large station radii and relatively small throw speeds this circle looks  almost as a straight line.

The object's trajectory is drawn in white. The display scale is recalculated for each new set of parameters so that the entire trajectory is always displayed on the screen.  The maximal absolute horizontal and vertical displacements from the point on the floor below the starting point (at the middle bottom edge of the display window) are displayed at the bottom of the applet's window.
You can see that for the default parameters, when the object is thrown vertically, its trajectory is not what we would expect on Earth (on Earth it should go up in straight line and then fall down following the same line). This is the effect of the apparent Coriolis force. You can see that the object travels about 3.5 meters up, but also about 0.66 m to the right.

Try changing the throw direction to  -7 degrees. Weird, isn't it? This would be definitely tough to reproduce under Earth's gravity. We have two almost straight lines with a sharp knee. You can imagine a very cool asymmetrical fountain on a space station - one jet going to the right in gentle and relatively "normal"  ballistic curve and another going to the left and forming this weird curve with sharp bend.  Would certainly turn some heads!

Now change the velocity to 0. What you see now is a trajectory of an object dropped from 1 m in a rotating space station. Again, the object does not fall vertically; its path curves slightly to the left and it lands about 4.4 cm left of the expected  impact point. This is the apparent Coriolis force at work. Actully, it is  easier to explain this behavior without resorting to Coriolis force. The simple explanation is that the linear velocity of the object at 1 m from the station floor is slightly smaller than the linear velocity of the floor (because the linear velocity is proportional to the distance from the center of rotation). So the object's motion to the right (in the dir. of the station rotation, perpendicular to its radius) is a little slower than the motion of the floor. The object thus "lags behing"  as it falls and lands slightly behind  (to the left of) the expected impact point when it reaches the floor. That's (almost) all  there is to it.

You can see that the deflection increases as the space station's radius decreases. Try  radius of 2 m - looks pretty serious, huh?

Interesting Examples

I encourage you to experiment with different settings. You can get really surprising curves for some combination of parameters, especially for objects thrown against the station's rotation. Try for example:

• velocity 91 m/s
• velocity direction: -81 degrees
• space station radius: 838 m.

Since the  velocity is large, you can see the entire station (red circle).  Check how the curve changes as you change the direction angle.

Another interesting set:

• velocity 44 m/s
• velocity direction: -25 degrees
• space station radius: 500 m.

Here you throw something to the left and then it comes around and hits you from the right. You can get more dramatic examples of this.

Another nice one:

• velocity 65 m/s
• velocity direction: -70 degrees
• space station radius: 500 m.

Almost orbiting forewer:

• velocity 70 m/s
• velocity direction: -89 degrees
• space station radius: 500 m.

One more:

• velocity 2 m/s
• velocity direction: -71 degrees
• space station radius: 2 m.

These are just a few examples. There are many more interesting combinations.

The Applet

The applet was written using Swing  (in Sun's Forte for Java IDE) and - as far as I can tell - it requires Java 1.2.2 or higher (Java 2). I tested it, it works fine on my PC, but I could not make it run on Java-crippled Macs  (why don't they have the current version of Java on the Macs???) or under IRIX. Well, hard cheese. I hope you can  make it work somehow; it should be possible to download all the .class files from http://vislab.cs.wright.edu/~rbryll/CORIOLIS/ and run the thing in an applet viewer if your browser cannot run it. Or you could also try to download the source .java file (a really short program indeed), compile it and run on your machine. I'm not a Java expert, unfortunately, so I cannot help you too much. The .html wrapper for the applet  was taken from Bruce Eckel's "Thinking in Java" - an excellent book, and available for free download (includes source code for all examples) here! The .html version of this book is extremely convenient to use, because searching is very easy (there is an index and also the browser's "Find..." command can be used).

I hope you enjoy the simulation (assuming that your browser runs it...).

Here it goes:

How about a Foulcault Pendulum?

After writing the above simulation I started wondering how a Foulcault pendulum would behave on a rotating space station. Now THIS would be a tempting  thing to simulate, but the simulation would be much more complex than what you see above. I would attempt it if I had time. Unfortunately, I cannot afford  it if I want to graduate before turning 60... My guess is that the pendulum would do wild things trying to preserve its plane of motion. If it was initially swinging along the station's axis of rotation, would its plane of motion (initially vertical, or along the station's radius)  - fixed in absolute frame of reference - deviate stronger and stronger from vertical, until the pendulum swings horizontally (when observed from the station's floor)? I guess not, this would be too crazy. It would be extremely interesting to see what the pendulum's trajectory looks like. Some kind  of perturbed 8-shape? If you know about an existing simulation of a Foulcault Pendulum in a rotating frame of reference, or if you have your own theories about it, PLEASE email me!!!! Thanks!