I’m more than a little bit edgy about the quarter-final against France tomorrow morning, so this post is going to concentrate on a couple of distinctive aspects of Forsyth-Barr Stadium in Dunedin, which is just about as far away from Cardiff as possible.* It’s also a bit of a follow-up to the previous post on kicking.

Forsyth-Barr Stadium (completed August 2011) has a roof, which comes in pretty handy during the Otago winter. When a stadium has a roof, the first thing that springs to mind (for me, anyway) is whether the ball can be kicked high enough to hit the roof. If you’re ever out on a field under a roof holding a rugby ball, hitting the roof is probably the first thing you’d try.

We can use kinematic equations to figure out the trajectory of your kick. If the ball goes straight up with an initial velocity (v), we know that an gravitational acceleration (g) acts straight down on the ball, so the vertical height it will reach before stopping (h) is given by

Now, the roof of Forsyth-Barr is on a bit of a slope, and the height near the edge of the field is reportedly around 30 m. Using the equation, we find that the ball has to be travelling at 24.3 metres per second when it leaves your boot if it is to hit the roof. How fast is this? In rugby terms, we might think about how far the same kick might go when it is directed downfield. You can figure out that if you kick the ball at some angle to the ground (x), then it will follow a parabolic arc, and the maximum horizontal distance travelled (d) is given by d = v² / g (which is the same as 2h in this case). For this maximum distance, x = 45 degrees.

So, we find that the kick that would reach the roof of Forsyth-Barr Stadium could travel 60 m on the full if kicked downfield. That’s a very long kick indeed, although I don’t think it’s impossible. To be fair to the Stadium designers, the roof height near the middle of the field is closer to 37 m, so the kicker would need to be able to kick the ball 74 m downfield, on the full, to be able to hit the roof at this point. The stadium’s website reckons that the highest observed kick of a rugby ball is 29.4 m.**

[If you want a harder trajectories problem, try adding some wind speed or some posts that need to be cleared.]

We’re not finished with Dunedin yet, because I’d like to draw your attention to a conundrum for goal-kickers at Forsyth-Barr. Shortly after the Stadium was built, and during the 2011 World Cup, it was noticed that goal-kickers weren’t having a great time in Dunedin. Internationally renowned aces Johnny Wilkinson (England) and Morne Steyn (South Africa) had sub-par outings there, and overall statistics were a little low.

In 2011, Brian Wilkins from NIWA in Wellington gave an explanation in press reports, suggesting that any swerve on the ball due to spin is accentuated by the roof because the air is relatively still, and non-turbulent. This explanation is based on the Magnus Effect, as discussed in the previous RWC Physics post. There are a couple of other ideas to throw into the mix:

1. The knuckleball effect, which takes its name from baseball, but is also used in (for example) soccer. The idea is that, if the spin on the ball is minimal, then different flows over different parts of the ball cause swerve (as in the Magnus Effect), but that the direction of swerve is unpredictable and erratic. Knuckleballs will be different for different types of balls (here is a video of some guys comparing different soccer balls against some fairly average goalkeeping) because the flow over the seams of the ball is important. The properties of the air should also make knuckleballs more or less prominent.

Erratic motion is great if you’re trying to throw a strike or beat the goalie, but not so good for a goal-kicker. From a rugby point of view, different types of kicker may be more (or less) affected. Morne Steyn, an excellent goalkicker, learned his trade on the high veldt in South Africa, where the atmosphere is thin and the ball flies straight and true. It’s quite possible that this kicking style meant that he was vulnerable to knuckleball effects at Forsyth-Barr.

2. Kickers get to practice on the field, with the roof on, before the match. Also, similar low kicking percentages aren’t observed at other covered stadiums. This leads me to think that the aerodynamics might be affected by changes in air temperature or humidity that only happen once a big crowd turns up in Dunedin.

*Oh OK you can think about Cardiff if you want: the Millenium Stadium roof is 33 m high.

**Here we’ve ignored that the ball is kicked from a little way above the ground, and that there are aerodynamic advantages to spiral kicks as discussed previously.