Thursday, May 04, 2006

Kicking in rugby (mathematical calculations and models are omitted as a result of the format of the blog)

There are four forms of kick used in rugby union, the punt, the drop-kick the place kick and the fly kick or dribble.

The Punt
The punt is a kick in which the ball is dropped from the players hands and is then struck by the foot before it hits the ground. It comes in a variety of forms used in the different phases of the game and in the different forms of oval ball football.

Drop Punt
This is used in the AFL in which the ball is struck so that it rotates end over end about a transverse axis.

Spiral punt.
The ball is struck eccentrically to the longitudinal axis so that it spins about the longitudinal axis and travels substantially in the direction of the longitudinal axis.

Banana kick.
This is a spiral punt in which the longitudinal axis attacks the air at a substantial angle so that aerodynamic forces cause the direction of travel to change markedly during flight.

Chip kick
This is a short range punt which is used to clear and turn the defender allowing the attacker to regather. It tends to have little rotation so that the initial bounce is reasonably predictable.

Grubber
The ball is stabbed downward into the ground so that it travels forward along the ground with substantial end over end rotation. The shape and rotation cause the ball to maintain forward motion while at times bouncing up where the attacker can regather it easily.

Drop Kick

The drop kick is similar to the punt except that the ball must first hit the ground and is then struck on the half-volley or when the ball begins to rise.

Place kick

The ball is stationary and held in position on some support, formerly a heel mark or earth mound, latterly a sand mound or a prepared plastic kicking tee.

To score from a kick or to restart the game by a kick it must be kicked from the ground, that is to say, a place kick or drop kick. For speed, the drop kick is the only kick used for a restart.


The Physics of the Kick

The rugby ball is a prolate spheroid having:
Mass = 0.44kg
Minor axis = 0.19m
Major axis= 0.29m
Moment of inertia about longitudinal axis approximately 0.002 kgm2
Moment of inertia about transverse axis approximately 0.008 kgm2

In the simplest analysis, the kick may be considered as a collision between a moving object (the foot and associated leg segments) and a substantially stationary object (the ball). If the collision is of relatively short duration momentum is conserved and the ball will gain momentum while the foot loses an equal amount of momentum.
If the leg is localised to the foot we get a ratio of about 3:1 for the masses assuming the kicker has a mass of about 80kg (body segment masses from Hay).
Analyses of a small number of touch kicks in junior elite rugby using digital video running at 50 fields per second were unable to detect a change of speed of the foot much over1.5 ms-1 when the trend in velocity is extrapolated through the contact zone.
This assumes that the foot is free or substantially so, which of course it is not. We can replace the foot in this analysis by one or more leg segments and consider the centres of mass. To do this we would need to assume that the leg is rigid at the instant of impact but moving freely about the hip, again neither of those assumptions is true and determination of the change of motion of the centre of mass using 2-D analysis at 50fps does not give reliable data.
Observation suggests that for longer kicks the knee is at or near full extension so that we could treat this as a collision between a rotating object (the leg) and a stationary mass (the ball).
Using the equations of angular momentum for a rigid body and a moving particle we can use the speed of the foot alone without needing to track the centre of mass of the leg in calculating the speed of the ball.
Taking the leg segment moments of inertia as given in Hay and applying the parallel axis theorem we can get an approximate value for the moment of inertia and this analysis gives a value of 8.6:1 which is a closer match to the data obtained above.
In controlled trials in American football punting, (Ryan D. Hartschuh
Physics Department, The College of Wooster, Wooster, Ohio 44691)
the foot was seen to be travelling at 16 ms-1 before and after impact. Extrapolating the motion before and after impact suggests a peak speed of about 19 ms-1 and a difference between peak speed and post impact speed of about 3 ms-1 . From knee angle studies (Orchard et al) of kicks in the AFL it would appear that at the time of contact the knee is usually fully extended rather than moving into extension and this is supported by our own observations of rugby. ( note (orchard 2) in 40m drop punts in the AFL it was noted that the knee was still flexed by 50o at the instant of contact) Thus the main effect should be a collision as above rather than an accelerated muscular action. We deduce, therefore, for a change in ball speed of about 30 ms-1 and allowing for the fact that the ball is dropping at 4 ms-1 before contact we get a final speed of about 26 ms-1. The studies above gave values of 24 – 25 ms-1.

There is general agreement in the literature that the speed of the ball is related to the speed of the foot at contact. This in turn is related to the angular velocity of the shank about the knee and the hip. Initially the support foot is placed to give substantial rotation of the hips about the vertical axis. The hips then rotate as the thigh is accelerated about the hip. To maximise this angular acceleration the knee is flexed reducing the moment of inertia of the leg as a whole. In the partially flexed position used by a rugby kicker, the moment of inertia of the leg in this position is about 1.7 kgm2 as compared to 3.8 kgm2 fully extended. Since , when I is less than half, angular acceleration will double giving a larger angular velocity as the knee starts to extend allowing the knee to gain greater velocity in the momentum transfer from the thigh to the shank. When the knee is below the hip the shank is accelerated and as a reaction the thigh decelerates so that at the instant of impact the pelvis has completed rotation, thigh and the knee have come to rest and the knee reaches full extension. There is some evidence that the support leg contributes to the motion immediately before contact but this requires further study.
If this is the case it would seem that at impact the moving body is the shank and foot system rather than the leg as a whole. Performing the angular analysis on these segments alone we get a calculated speed ratio of 5.5 which suggests that there must be some contribution from the thigh and the pelvis. In the events that we have analysed the speed of the foot is a maximum at the instant of impact.
In the follow through the knee moves into slight hyperextension, the thigh accelerates and the shank decelerates. It has been observed that kickers will lock the knee on impact, tensing the muscles of the leg so that it behaves as a rigid body with a larger moment of inertia during the collision through the larger effective mass or inertia. Some kickers will actually rise off the ground suggesting support leg driveduring the kick.

The time of impact has been measured as between 10 and 15ms (less than one field of digital video). From this we can deduce the force on the ball or foot (they have the same magnitude) by Newton’s third law of motion : action = -reaction.

Using the figures above this gives a calculated force of 1015N

High speed video of a dry ball gave an average force of 1030N with a softer ball and 1020N with a harder ball (Orchard).

In the punt the angle of release (vertical elevation) is determined by the position within the leg swing at which the foot makes contact with the ball and the direction of motion of the foot at contact. This can also be modified by the upper body lean at the time of contact so that the direction of foot movement is altered relative to the phase of the leg swing. By leaning forward and placing the body over the ball, the foot is moving at a much smaller angle to the ground and the flight will be much flatter. In an extreme case the ball may be struck before the knee is extended and the foot is travelling downward so that the ball is struck onto the ground resulting in a grubber kick. With a greater backward lean and contacting the ball in front of the body, the elevation is much more nearly vertical and the kick becomes a “bomb”, “Garry Owen” or “up and under”.

The drop kick is similar to the punt in that the ball is moving and is in the air (or at least moving into the air) when it is struck. Since the ball is close to the ground, the foot must also be relatively close to the ground and near the lowest point of its arc. The velocity (direction) of the kicking foot is controlled by the placement of the support foot and the body lean at contact. The ball, however, has an upward component of motion (of about 3ms-1) when struck giving a greater angle of release than for an equivalent punt.
The total momentum of the ball and “foot” before contact is greater for the drop kick than for the punt as both are moving in the same, as opposed to opposite, directions. The total energy of the system in the drop kick is slightly less due to the loss of energy on rebound, but the difference is insignificant. Assuming the same coefficient of restitution, we can show that more of the energy is retained by the foot to give the same total energy but less momentum after the kick and therefore the ball must travel slower in a drop kick than in the equivalent punt. This results from the shorter time of contact because both objects are travelling in the same direction and the ball is accelerated from 3ms-1 to exceed the foot speed in less time than it takes to accelerate the ball from -4ms-1 to exceed the foot speed.

The place kick (awaiting data)

The ball in flight

First we consider the ball as if it were a simple projectile (i.e. assuming no air resistance, no spin and ball behaving as a smooth sphere).
In this case the release height where the foot strikes the ball is substantially identical to the landing height (whether it is caught or not) so that the path may be considered as symmetrical about the apex of the trajectory. The horizontal component of the velocity is constant (Newton’s first law of motion – in the absence of a horizontal force the object continues in uniform motion). The vertical component is subject to gravity and has a downward acceleration of 9.8ms-2.
The motion can be solved completely using the kinematic equations for motion with a constant acceleration in a straight line.

Using the standard kinematic equations we can derive expressions for the height and range of the ball.

(Hertschuh) reports that a real football (American) travels 24% to 33% less than calculated, in kicks of 40m to 50m actual travel, as a result of air resistance. The flight time was, however, slightly greater than that calculated.

Applying the formula to calculate the drag on the ball at each instant it is possible to obtain a good approximation to the range and time of flight in real kicks. The drag coefficient is not constant in higher speed kicks, but drops suddenly as the speed exceeds a critical value (about 15ms-1). The model used accommodates that change as a stepwise decrease at 15 ms-1 speed magnitude. Accordingly the true trajectory is not a parabola but becomes much steeper in the descending portion than in the ascending portion. Calculations have been done for a ball travelling end on but the drag would be substantially larger for a ball tumbling in flight and thus presenting a larger average frontal area to the air flow. These give a correct trajectory and approximately correct range (slightly low) but not the extended flight time.
Spin of the ball about the longitudinal axis, as in the spiral punt, stabilises the attitude of the ball so that the frontal area will tend to increase as the angle between the long axis and the direction of travel increases during the flight. This will tend to accentuate the increase in drag due to increase in drag coefficient in the latter part of the trajectory.
The larger angle of attack in the second half of the trajectory will, however, give some lift so that the time of flight will be increased.
If the spin of the ball were about the line of flight at all times this would have no effect on the speed of air flow over opposite sides of the ball and the pressure and thus force on each side would be the same. Any angle of attack, however, will give a slight speed differential between the two sides causing a pressure difference and a force towards the side having the larger relative air speed. This is called the Magnus effect. This effect can be enhanced if the axis of the ball is not parallel to the vertical plane including the velocity to give a banana kick. Since the effect of this force becomes greater as the velocity of the ball decreases, a kicker may gain greater distance with a touch kick by kicking substantially parallel to the touch line to allow the transverse force to take the ball in to touch late in the flight.
Causing the ball to tumble end over end, as in the drop punt in AFL, reduces the range of the kick as a result of the greater air resistance. It does, however, give greater accuracy, because of the lack of transverse aerodynamic forces, and is easier to catch so is useful as a chip kick or cross kick to a winger. The tumbling action also means that when the ball hits the ground it will tend to roll end over end for some distance. Having a large moment of inertia, it also has a substantial rotational kinetic energy to maintain its rotation as it moves forward. This is frequently exchanged with translational kinetic energy as the ball bounces up or forward randomly depending on the attitude of the ball as it strikes the ground. The patient chaser is thus rewarded as the ball eventually bounces to a good catching height while continuing to move forward predictably.