tits&tans strides on to the pitch for the Titans in a scholarly fashion, wearing a tweed jacket, sucking on a pipe and adjusts his monocle.. then realises this is not the right place for dress up ...
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Trajectory of a prolate ellipsoid: perturbations in 3D gravitational forcefield
Being derived from football, Rugby League (RL) involves a large amount of kicking, both from set pieces and in general play. It is an important skill to master and often results in last-minute success.
2 Mechanics of Kicking
2.1 Types of Kick
2.1.1 Place
Used to start the game at kick-off, to convert tries and/or to kick penalties. The ball remains stationary and fixed in position on some support (a heel mark, sand mound or kicking tee).
2.1.2 Punt
Often used as an attacking manoeuvre, the bomb can create confusion amongst the opposition’s defence. The ball is dropped from the player’s hands and struck by the foot before it hits the ground.
2.1.3 Drop
Used to restart a game from a drop-out under the posts or to kick for a field goal. Similar to a punt, except for that the ball must first hit the ground before being struck.
2.1.4 Derived Kicks
Hybrid kicks abound in this sport, including the chip, banana, torpedo and the grubber.
2.2 Biomechanical Process
RL kicks follow some version of this sequence:
- Plant
- Upper limb acceleration
- Lower limb acceleration
- Contact
- Follow through
3 Background
3.1 Ellipsoid's Speed
It is known that the speed of the ball is related to the speed of the foot at contact. However, the ratio of foot to ellipsoid speed, while less than unity, is still open to further research. Foot speed is related to the angular velocity of the knee, the rotation of the pelvis and the forward motion of the player.
3.2 Round-the-corner Style
Rather than the traditional “toe-poke”, Willie Home developed the round-the-corner style in the 1940s. This was imported into Australia in 1975 by John Gray, playing for the North Sydney Bears. This style is considered superior due to:
a) A increased chance of middling the ball.
b) An increased foot extension, leading to higher foot speed and better momentum transfer.
3.3 Upper Limb Swinging
The swinging of the non-kicking-side arm across the chest, upon contact with the ball, has been shown to improve the accuracy of a long distance kick. The added momentum from this swing helps control the pelvic rotation and counteracts the leg movement, both of which reduce the margin of error.
4 Trajectory Calculation
4.1 Assumptions
- A simple (spherical) projectile (no air resistance or spin).
- Symmetrical trajectory about the apex.
- Horizontal velocity is constant.
- Vertical velocity is subject to gravity (downward acceleration of 9.8ms^-2).
Standard kinematic equations for motion in a straight line with constant acceleration, using SI symbols, resolves the velocity:
Horizontal
s = 2utcosθ Vertical
v = usinθ – gt
At the apex
v = 0. After further substitution, this gives:
t = (u^2)sin2θ( g^-1).
It is now possible to derive an expression for the maximum height the ball reaches. The shape of such a trajectory is a parabola.
4.2 Air Resistance
Calculating the drag (force) on the ball each instant involves knowing the drag coefficient and altering the above acceleration accordingly. The trajectory is now steeper in the descending portion than in the ascending portion. The basic assumption made here is that the ball is not tumbling in flight and presents a minimal frontal area to the air flow.
4.3 Spin
If the ball spins about the longitudinal axis, the frontal area increases during flight. This raises the drag coefficient, and hence magnifies the drag on the ball. In turn, this provides lift and lengthens the ball’s flight time.
If the ball spins about the line of flight at all times, there is no effect on the speed of the air flow and thus no noticeable difference to the trajectory. Should the air flow, however, be unequal about the ball, the Magnus effect comes into play, and the trajectory becomes unstable and increasingly unpredictable.
Causing the ball to tumble end over end reduces the range of the kick, but does give greater accuracy.
5 Conclusion
Although it has been demonstrated that, in theory, a trajectory is straightforward to predict, it’s unlikely that Jonathan Thurston or even the legendary Daryl Halligan took the time to study such prolate ellipsoidal paths in such detail.
-v-
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