Notes on the Freefall Trajectory Simulation

John S. Kallend PhD, D-23151

These programs attempt to predict the MOST LIKELY trajectory of a skydiver leaving a jump plane at 14,000 feet and freefalling to 3,000 feet. Two cases are considered - one, a RW skydiver (or group of skydivers) who flies in the high drag "belly to Earth" position, and the other, a freeflier (or group) who adopts a low drag flying position. In each case, they are assumed to maintain the same drag coefficient for the entire jump. (In other words, they maintain the same effective position with respect to the relative wind - initially the relative wind will come at them horizontally due to the jump plane's forward motion, and gradually this will transition to a vertical direction as they dissipate their forward speed and gain vertical speed).

Another way of looking at this is to consider the trajectory to be that of Olav Zipser's "skyball" that has been designed to fall at the appropriate speed for the discipline - it simply follows the ballistic path without any steering or fall rate control being exerted once it has been released from the plane. The skyball defines the ideal path, with no sliding, funneling or corking.

There are several contributions to the trajectory: the forward throw due to the speed of the jump plane, drift in the wind, passing through wind shears in the atmosphere, and any systematic movements by the skydiver (backsliding, tracking etc.)

I excluded anything that involved the vagaries of human behavior (hence the "skyball" analogy). The two cases I consider could be considered as extremes. I don't think a belly flier will go less far in forward throw than my calculation, and I don't think a head downer will go farther unless they make a conscious effort to do so. Someone who 'mixes it up' will go some intermediate distance. Most of the RW fliers I know try to maintain an essentially constant attitude with respect to the wind, and probably achieve this unless they funnel. I suspect the VRW flyers may vary rather more unless they reference their position to a skyball. I don't freefly, so I asked Tamara Koyn for her perspective. Her notes are printed at the end of this page. Also check out Bryan Burke's comments on exit order.

Excluded from the 2-d model are such things as unintended backsliding, freaking around, funneling, corking etc. I believe these will impose a random spread on the calculated paths. A typical group will achieve separation by tracking at the end of the dive - the path plotted by the program will represent the center of the group; individuals may have separated from this by 500 feet or more during the their track. The 3-d program shows a "cone of uncertainty" corresponding to random sliding, which may be adjusted by the user.

How Does It Work? (Nerd Stuff)

It's just f = ma (Newton's Law). The forces acting are gravity and aerodynamic drag. At skydiving speeds the Reynolds number (R) of a human body is around 500,000 or so, where drag varies with the square of airspeed, and this is the drag law that has been assumed. Unfortunately this non-linear drag law introduces a coupling between the horizontal and vertical components of the trajectory, which complicates the solution somewhat. The drag coefficients for the two cases were chosen to give a 50 second freefall for the fast faller, and a 65 second freefall for the slow faller (These are adjustable in the 3-d program). For more information on drag, I highly recommend Hoerner's book "Fluid Dynamic Drag".

Atmospheric density variation with altitude was calculated according to the ICAO standard atmosphere. A simple 2-level wind profile was assumed in the 2-d program, and more levels in the 3-d program.

The resulting system of differential equations cannot be solved analytically (well, not by me, anyway). I used a 2nd order modified Euler's method numerical solution with a time increment of 0.01 seconds. I played around some and found no advantage with a 4th order solution or with a smaller time slice. All calculations are in double precision to minimize accumulation of rounding errors (although, once again, the equations are sufficiently well behaved that this is not really a problem).

Some observations:

Because they fly with higher drag, RW fliers will more quickly dissipate their forward velocity inherited from the jump plane than VRW fliers. The net result is that the RW flier gets less distance in "throw" down the line of flight. The difference in throw between the two cases is equivalent to the distance flown by the jump plane in around 3.5 seconds, regardless of jump plane speed.

The faster faller drifts with the wind at the same horizontal speed as the slow faller, but because his/her vertical speed is higher, will have a steeper trajectory.

If a skydiver passes through a wind shear, the trajectory takes about 2000ft to come to equilibrium with the new wind speed. In other words, the drift due to the upper winds persists for several seconds longer than the time the skydiver is actually in them.

Unintended deployment may occur at any altitude - any scenario where trajectories cross is potentially dangerous.

Don't forget that once the canopy is open, drift with any wind at opening altitude will superimpose on the canopy flight path.


This tool is offered to help you draw your own conclusions. Decisions about exit order and exit spacing depend on more than just physics. Experiment with various scenarios and discuss with an experienced skydiver.

Some notes on freeflying from Tamara Koyn

Regarding exit order, I'd like to share a few comments from my personal perspective of being a freestylist and a freeflyer.

Good freeflyers who are doing truly Vertical Relative Work are also falling relatively straight down. When working with my students on relative control exercises, I enforce that. In this manner, no accidental sliding will cause us to move over into the airspace of an RW group and crash through them when exiting afterwards. We are always taking at least 8 seconds to exit the plane. When exiting nearly last in practicing freestyle or freeflying which I have done for most of my entire 2,500 jumps since 1985, I have found that the distance between myself or my freefly group and the others to be large. When it has been demanded that I exit first, I always found others uncomfortably close to me, a number of close calls that cause bad dreams, and a landing accident.

Novice freeflyers can move very dramatically while learning the basic head-down or headup flying stance. These novices should do several things to ensure their safety and the safety of everyone on the jumprun. Firstly, they should face 90 degrees off the line of flight while they practice a position. Any horizontal motion will carry them off the line of flight. Secondly, they should not hold positions for long periods of time. They should be practicing their transitioning skills as well. If they are unable to note the direction of the jumprun and face 90 degrees off the line of flight or resume facing 90 degrees off the line of flight any time after getting out of control, perhaps, they are not yet ready to begin learning freeflying. (If the novice is taking lessons from my freefly program, they have specific instructions for each training dive they do, ensuring that there is no mishap in the airspace of the jumprun.)

Additionally, it is to a novice's benefit to open a little higher as well. This helps keep vertical separation of canopies for landing within the target area. Drop Zones that have tighter landing areas may wish to require freeflyers to open higher so as to avoid congestion in the landing area. Freeflyers should also consider landing in the student landing zone if that is what is necessary to maintain safe separation.

Freeflyers like to say that freeflying is free and they also include tracking activities under the heading of freeflying. They should not. They should call it by its real name, tracking or tracking RW. Their behaviors on jumprun should be different. Usually, it is to exit last and to diverge away from the jumprun doing their tracking activities. When correctly planned, they will end up far away from the standard string of jumpers from the jumprun. If there will be two tracking groups on a single jumprun, they will have to communicate with one another so that they will take on diverging pathways both diverging from the regular jumprun.

So those are some of my comments...

I welcome you to stop by my web site at Clouddancer

Of particular interest, you will find information from the 1999 Quincy Seminar on Spotting and Separation: .

Blue Skies and Keep Safe Separation!
Cloud Dancer

JK 8/99