Are Figure Skaters Projectiles?

The title above is eye-catching on a number of levels.  Have you ever been on those high sessions with 25 skaters?

Actually, this post is a follow-up to my discussion of slo-motion versus computer analysis.  In that post I made the bold claim that the minimum height for a double axel is 60% higher than the minimum height for a double lutz.  In this post, I’ll show how I got those numbers.

Warning:  This post is extremely technical so if you just want the final answer, see the height and flight time tables below.

The work of figure skating video analysis experts has demonstrated that there is a minimum flight time requirement for each of the jumps.  For example, a double lutz needs to be in the air for 0.367 seconds or longer while a double axel needs to be in the air for 0.467 seconds or longer.  These values have been determined by analyzing hundreds of jumps with various video analysis programs such as Dartfish or Pro-Trainer.

Using the video analysis software, a coach can count the number of frames a skater is in the air.  You do this by advancing the jump entry and take-off frame-by-frame until the skate blade just leaves the ice.  You mark that as your starting point and advance the jump frame-by-frame through the flight until the skate blade just touches the ice.  The number of frames from the start to the end point give the flight time, as video cameras shoot frames at fixed time intervals.

In North America, the video standard is 29.97 frames per second but the magic of modern video analysis software such as Dartfish and Pro-Trainer allows those programs to double the number of pictures for certain consumer video cameras.  (In a future post I’ll discuss the technical details of interlace vs. progressive scan.)  That means we can resolve the jump flight time to 1/60th of a second (actually 1/59.94 of a second).

So here’s an example.  I’ve never analyzed a double axel with less than 28 frames of flight time at 60 frames per second.

28/60 = 0.467 seconds.  Dartfish and Pro-Trainer have timers so you don’t actually have to count the frames.

Now to calculate an estimate of how high the jump was, I apply the laws of physics.  In basic physics courses, there is almost always a part of the course devoted to “projectile motion” and the associated equations.  We’re going to consider our in-flight skaters as projectiles!  For our situation, the equation of interest is the simplified time-acceleration-distance equation. 

This equation states that the distance an object travels under constant acceleration from rest is one half the acceleration rate times the square of the time.  The equation looks like:  Distance = 1/2 x Acceleration x Time x Time.  For our skating jump analysis, we need to use half the total flight time as the Time in the equation as the skater at the peak of the jump has no vertical speed (“from rest” at the top of the jump to full speed at landing… to use the simplified equation).

The acceleration of a projectile is simply the pull of gravity.  And gravity has a constant acceleration of 386.088 inches/sec2.  Using the double axel minimum flight time of 0.467,  the decent-only time is 0.467/2 = .2335 seconds.  Plugging all of this into the equation yields:

Distance = 0.5 x 386.088 x .2335 x .2335 = 10.53 inches

Just for theoretical completeness, the actual jump height is slightly lower.  The reason is that a skater always points his or her toe at take-off but usually flexes the toe in the air and upon landing.  This makes the number of frames method a tiny bit inaccurate.  But the result is close to the theoretical.  A double axel must be 10.5 inches high or you can forget it.

(Some skaters land with their landing leg slightly bent.  Their jumps are actually slightly smaller than that indicated by the table below.)

Here’s the whole table:

Frames in Air	Flight Time in Seconds	Height in Inches
10	0.1667	1.3
11	0.1833	1.6
12	0.2000	1.9
13	0.2167	2.3
14	0.2333	2.6
15	0.2500	3.0
16	0.2667	3.4
17	0.2833	3.9
18	0.3000	4.3
19	0.3167	4.8
20	0.3333	5.4
21	0.3500	5.9
22	0.3667	6.5
23	0.3833	7.1
24	0.4000	7.7
25	0.4167	8.4
26	0.4333	9.1
27	0.4500	9.8
28	0.4667	10.5
29	0.4833	11.3
30	0.5000	12.1
31	0.5167	12.9
32	0.5333	13.7
33	0.5500	14.6
34	0.5667	15.5
35	0.5833	16.4
36	0.6000	17.4
37	0.6167	18.4
38	0.6333	19.4
39	0.6500	20.4
40	0.6667	21.4

It’s pretty fascinating to understand the ramifications of this table.  For example

• A double lutz needs to be have a miminum flight time of about 0.36 seconds, so it will be 6.5 inches high. 
• If a triple lutz needs to have a minimum flight time of 0.58 seconds (estimated), it will be 16.4 inches high.

OK…has that sunk in yet???  The triple lutz needs to be 250% higher than the double!!  WOW!  Do you see why so few skaters actually get all those triples?

I hope this was interesting and useful.  If you are enjoying this blog, please pass the URL along to your friends that may be interested.  Also, please leave a comment as I would love to hear from you.

Trevor