Wednesday, May 31, 2006

The Scrum, the essence of rugby
Football as a formal contest between sides developed in the English public schools in the early 19th century. Each school developed its own traditions rather than formal rules since they are agreements between children in the playground. Similar well governed contests are still to be seen in schools everywhere where younger children learn the rules from older children and local rules apply. When these games were confined to intramural competition this was fine and no written code was required. To avoid disputes, rulings would occasionally be made on marginal events, but rarely was a full description written down since the players knew by observation what was required. Hence the main features of many of the games are obscure. Some pictures do exist of games being played.
In many of the games the main aim was to propel the ball towards a goal which was often the entire opponents' end of the field or could be some restricted part of it. This was usually done by kicking, but some schools allowed the ball to be struck by the hand. The numbers of players was often unrestricted so that large numbers would chase the ball and form a large melee kicking and hacking blindly trying to drive the ball through the opposition towards the goal. Rules were developed to limit the damage but hacking an opponent's shins was an integral part of the game. This mass of standing players was called ruck or scrummage.
In some versions of the game including that played at Rugby school, a player was permitted to catch a moving ball before placing it to kick or for a colleague to kick to resume the game or running back toward their goal (this was of advantage with certain offside rules). When the ball went dead it was then replaced on the ground between the first players of each side to arrive who then scrummaged the ball. The remnants of this exist in the set scrum and lineout.
In th1820s0s players at Rugby began to run forward carrying the ball forward rather than kicking or running back (which had been permitted). Later this change was attributed to William Webb Ellis, but this is almost certainlapocryphalll Opponents were allowed to attempt to challenge the carrier by holding him in a maul or hacking him.
A number of written rules were produced without ever describing the main features of actual games of which scrummaging was an essential part. Eventually competitions were arranged between schools and more particularly between old boys at university and rules had to be agreed before play commenced. The Cambridge rules of 1863 were one such attempt, but these excluded many of the features of the Rugby game and other groups formulated the laws of the Rugby game separately producing the split between "association" football (the soccers) and Rugby football (rugger).
Reduction in numbers led to a more open game with fewer players in the scrummage (forwards) and similar numbers standing off (backs). The restart scrummage became more formal with equal numbers in reasonably fixed positions. The laws on scrum, ruck and maul have evolved in response to tactical changes by teams and the needs of safety. For example, the 1905 New Zealand team developed a two player front row that destabilised the scrum (and brought them great success) so the number has since been set at three no more no less. They also introduced wing forwards who tracked the offside line ( at that stage the ball) and hence stifled the opposition back play. No matter that modern backs consider the scrum (loose or tight) merely as a way of getting those lumbering forwards away from the real play, it is clear that the scrummage is the historical essence of rugby and it is the various forms of scrum that make rugby unique. All other forms of football have abandoned the scrum as a contest or have formalised it out of the game.
Crusaders do it again
I went to Jade stadium on Saturday to join the crowd celebrating the Crusadrers winning the Super 14. I can't say I went to watch them as the fog hid any play beyond the 15m line on our side and even if you could see the players, the ball merged into the mist so you just had to guess what was happening by watching the near wingers running or listening to the crowd nearest the play.

Thursday, May 11, 2006

physics of the tackle

In this preliminary analysis we will consider the tackle according to:
1 Relative direction of ball carrier and tackler (front, side or rear)
2 Height of tackle (legs, Mid below com, mid above com)

For the first stage the ball carrier is assumed to run upright and to behave as a rigid body during the tackle (flexibility and the changes it causes will be considered later). The tackler will be assumed to strike the carrier at a single point and grasps the ball carrier so that they behave as a single object after the tackle. The players are also assumed to stop running on impact.

A) Tackle from in front on the lower legs. (It is difficult to make this tackle).

Torque and angular acceleration: The force exerted on the point of contact makes a small angle with the horizontal and is a relatively large distance from the centre of mass of the carrier thus exerting a large torque about the centre of mass. Relative to the centre of mass of the carrier, therefore, there will be a large angular acceleration of the feet backward, that is, the carrier will rotate forward about the feet. The higher the tackle is made, the smaller the horizontal component of the force and the shorter the distance to the centre of mass.
Once the centre of mass has moved forward of the base, or because the carrier can no longer drive with the feet and loses dynamic equilibrium, the weight of the ball carrier will exert a torque about the point of contact or feet and the rotation in the forward sense will accelerate at an increasing rate as the CofM moves forward. The carrier at this stage can no longer recover even if released and the tackler is able to release and recover a standing position more quickly.
Angular momentum: Viewed from any fixed point or axis, a moving mass has angular momentum about that axis. This angular momentum is conserved if any force exerted on the mass is acting along the line joining the mass to the axis. Viewed relative to the transverse and frontal axes, once the tackler has grasped the carrier, the force between them becomes tension in the legs, that is, along the line from tackler to centre of mass. The angular momentum of the running mass is then converted into rotation of the carrier about the tackler. The moving mass may be either the ball carrier approaching the tackler or vice versa or a combination of the two. In terms of the rotation caused the end result is the same and the ball carrier falls forward toward the ground.
Viewed about the vertical axis, a safe tackle will tend to exert a small eccentric torque causing rotation of the ball carrier towards his back and leaving the tackler face down ready to recover his feet quickly. While the torque is small, the moment of inertia about this axis is also small allowing for a large angular displacement.

Impulse and momentum: At the point of impact, the force exerted by the tackler on the carrier and the force exerted by the carrier on the tackler are equal and opposite (Newton’s third law) and the time of contact is the same for each. Assuming the tackle is taken as an isolated event, the impulse (F.t) will give rise to an equal and opposite change in momentum for each body. E.g. if the tackler is stationary and the carrier is moving the carrier will be slowed in the impact and the tackler will be carried backwards. On the other hand a fast approaching tackler can drive the ball carrier backward without using drive from the ground reaction. These motions will all be subsequently altered by ground contact forces. Since the change in momentum of each player is the same, the more massive player will change velocity by a smaller amount giving dominance to the larger player in the tackle. (The smaller player can of course accelerate more rapidly to avoid the tackle).

Energy and work:

Since energy depends on the square of the speed, a higher speed player will have much more energy than a lower speed or stationary player. In a tackle between two players moving at different speeds the final speed will be between the initial speeds of the players so that the faster must lose more energy than the slower gains. In general the tackler in slowing the carrier down, therefore, does more work on the carrier than vice versa.

B) Tackle from side on legs. (common tackle on and by outside backs)

This time the force exerted by the tackler has a lateral component as well as a component backward relative to the motion of the centre of mass. Thus the angular momentum of the carrier is both forward and across the tackler so that the carrier will rotate about both the transverse and frontal axes, and will fall forward and to one side. The loss of drive from the feet and the tackling force will result in a net torque from the weight of the carrier and the tackle force causing the same accelerated rotation as before.
The sum of the momenta of the two players will now have a lateral component so that they will together move towards the side away from the point of contact subject again to ground forces.

In both of the above tackles, the rotation of the ball carrier about the centre of mass may be sufficient that the feet are carried off the ground for a substantial period while the centre of mass is falling under gravity. If the tackler is able to retain his feet and thus continue the drive around one of the horizontal axes it is possible that the ball carrier will become inverted and the tackle will become a spear tackle. The tackler must, therefore, ensure that the drive stops and tackle is completed before the ball carrier is moved past the horizontal.

C) Tackle from behind on legs. (catching a slower player who has broken through).

Normally the relative velocity of the tackler and the ball carrier will be small so that the tackle depends on removing the forward drive from the feet by wrapping the legs and applying a relatively smaller backward frictional force on the tackler once the tackle has been made. As a consequence a net torque is applied forward relative to the feet (or backward below the cofm) so that the tackled player falls forward. It may be sufficient merely to prevent the drive momentarily as in an ankle tap since the inertia of the centre of mass will be ahead of the player’s base and he will be unstable. Without the forward component of the ground reaction force, the ball carrier will rotate forward to a position from which it is difficult if not impossible to recover equilibrium. In any case to recover equilibrium will require the leading foot to reach ahead of the body giving a braking reaction and slowing the player down. Occasionally the relative velocity may be large, for example, when a chaser meets a stationary catcher who is facing towards their own end of the field. In this situation the feet of the carrier are carried forward relative to the com and the net torque rotates the player backward to land on their back.

D) Tackle front on, near or below centre of mass (midriff).
This is the classic tackle taught to beginners and uses the “cheek to cheek” contact.

Torque and angular acceleration: The force exerted by the tackler who has adopted a low position with a straight back is now angled upwards at an intermediate angle to the horizontal and is directed close to the com so that the moment of the force about the com is small. If the ball carrier is relatively upright and the reaction of the ground has only a small horizontal component, the net torque on the tackler will be small. The low body position of the tackler means that the impact force will exert a substantial torque about the feet of the tackler causing the tackler to rotate backwards and upwards about the feet. If the tackler has a firm grip on the ball carrier this will cause the carrier to be lifted as the angle of the tackler to the horizontal increases.
The tackler will endeavour to grip the carrier firmly while allowing the grip to fall down the legs preventing the carrier from continuing to run. The high contact and descending grip has the advantage of avoiding the need to try to grip the moving knees yet allowing them to be held in completing the tackle.
To enable the tackler to avoid neck damage it is normal to aim the head to one side and engage the ball carrier with the shoulder (the cheek to cheek position). In this position the force of the tackler on the carrier is eccentric to the vertical axis of the carrier exerting a small net torque about the vertical axis and causing rotation of the carrier out of the direction of travel. This will tend to place the tackler on top and in the best position to recover his feet and secure possession. To facilitate this the tackler will move into a tuck position which reduces the moment of inertia allowing quicker rotation from the prone position at the end of the tackle to an upright position and placing the body in a strong crouch.
A ball carrier who anticipates the rotation in the tackle by commencing rotation before contact can alter the point of contact to his advantage and make use of the additional torque to spin out of the tackle


Momentum and Impulse:

Since the line of action of the tackling force is through or close to the centre of mass of the carrier the collision at the instant of the tackle can be considered as a pure linear collision with conservation of momentum. Thus a stationary or slow moving tackler will be driven backward in the tackle by a fast moving carrier at a speed determined by the original velocity and the masses of the players.
If the tackler has a greater mass and speed than the carrier, the carrier will be driven backward.

For equal masses:
the stationary tackler and the moving carrier will move at half the original speed and the impulse on each will be half the momentum.
Tackler and carrier at equal speeds will come to a halt and the impulse on each will be equal to the momentum of each before contact.
Moving tackler and stationary carrier is the reverse of the original case and the impulse is again half.
Other combinations of velocity and mass will give intermediate values of impulse.
The time of impact is determined by the distance over which the impact takes place so that a tackle aimed at a bony body part, e.g. hip, will take less time than one aimed at a softer area, e.g. midriff and the net force at the point of contact will accordingly be larger. The use of padding, as allowed by the laws, will not affect the overall impulse, but will give some increase in time and thus decrease in force with the aim of protecting the bony parts of the tackler e.g. the collar bone.

Tackle from the side or behind: Since there is no backward component in either case, the carrier will continue to move forward taking the tackler with them. For equal masses and speeds the pair will move diagonally at about two thirds of the speed and each will experience an impulse of half the original momentum. From behind the impulse will decrease, reaching zero when the speeds are equal. In the case of the side on tackle, there will still be some torque enabling the tackler to turn the carrier and again gain a dominant position on the ground.

Tackle above centre of mass
In front

Since the point of contact is high, the tackler must be relatively upright and the force exerted by the tackler from the ground will have only a small horizontal component. The moment arm is also relatively small as the point of contact cannot be above shoulder level. Accordingly the torque is relatively small about the centre of mass. If the carrier is, however, making good ground contact and/or is driving through the feet. The force of the tackle and the ground contact will act as a force couple giving a some torque in the backward sense about the feet forcing the carrier to rotate backward with the tackler on top. Often, however, the tackler will use his own weight to increase the inertia of the carrier’s upper body causing him to fall forward. In this case the tackler will need to twist during impact to gain the dominant position and will require a wide base to give sufficient torque by ground reaction.

From side

The results are similar to the previous case except that the torque is now both to the opposite side from the tackler and backward (from the ground contact). This will make it easier for the tackler to turn the carrier and gain dominance.


From behind

Now the tackler can exert almost no torque in the contact and must rely on ground contact forces after initiating the tackle to bring it to completion. It is not unknown for the ball carrier to ‘carry’ the tackler for a considerable distance in this position since a smaller tackler is unable to exert large forces on the ground while gripping the upper body of the carrier.

Additional factors affecting the tackle

The majority of the discussion above assumes that the ball carrier is a rigid upright body in contact with the ground only in so far as is necessary for running at normal speeds. The body is of course flexible with major relevant joints at the hip and the knee. Also by adjusting orientation during the tackle phase it is possible to vary the ground contact forces and thus the reaction thereto and the thrust acting on the ball carrier. The ball carrier has the ability to modify the point or direction of impact by changing the body orientation immediately before impact.
The tackler has been assumed to be a form of projectile exerting a force on the tackler only because of their relative motion at impact. By using the reaction to ground contact forces the tackler can adjust the speed of impact and can change the direction of the force by altering body orientation immediately before impact. The tackler can also extend the effect of the tackle after impact by maintaining leg drive during and after impact. They may also exert torsional forces on their own body and on the carrier to cause rotation during the tackle.

The ball carrier

By bending at the hips and using a bent knee running action the ball carrier will lower the centre of mass and reduce the distances of the extremities from the centre of mass. This will increase the stability of the carrier and will reduce the moment arm of the force exerted by the tackler. These will reduce the chance that the carrier will be rotated in the tackle. On the other hand, if the tackler is able to overbalance the carrier the smaller moment of inertia in the crouched position may result in a quicker grounding. The higher knee action in this style of running makes it more difficult for the tackler to grasp the legs and the ball carrier may be able to break the tackle by driving through, that is, to continue to run vigorously so that the tackler cannot complete the wrap.

The ball carrier can disrupt the tackle if the tackler is approaching from an arc to the side by leaning towards the tackler with the upper body so that the legs are angled away from the tackler. The weight of the carrier will, of course, exert a torque tending to make the carrier fall toward the tackler which must be counteracted by an inward (centripetal) drive through the legs, or by a fend exerted against the tackler. The tackler thus finds it more difficult to reach the target and the tackle is easier to break. To reach the ball carrier, the tackler must adopt a less stable low upper body or a forward lean and the downward fend in this case will increase the forward torque on the tackler who will then rotate or fall forward.

When tackled from a forward side arc, the tackler exerts an eccentric force at impact, causing a torque about a vertical axis through the carrier. By initiating rotation in the same sense the carrier can enhance the effect of that torque, moving the target area away from the point of impact into the weaker region of the tackler’s arms and thus rolling out of the tackle using the small moment of inertia about the vertical axis to aid his rotation. Lifting the arms and ball above the tackle not only frees the ball for a subsequent pass, it may also cause a reduction in moment of inertia for the ball carrier accelerating the spin out of the tackle. If this is not successful however, it has exposed a vulnerable area of the carrier to the tackler.

Just before impact in the tackle, the carrier can step forward into contact, (the ‘power step’ increasing the momentum of the carrier at impact and rotating the body to present the stronger side aspect to the tackler. This will also drop the centre of mass, and widen the base of the carrier at the time of impact so the carrier is more stable and thus less likely to be taken to ground in the tackle. The carrier also to some extent gains control of the height of impact and can exert sufficient torque on the tackler to initiate backward rotation. If the step is away from the core of the tackler, the impact will be on the arm which is weaker than the shoulder, it will exert a large torque on the tackler so that the momentum of the ball carrier will now be directed towards the hands of the tackler making the grip more difficult to maintain, ie, breaking the tackle.

The tackler

The tackler will normally maintain full contact with the ground and leaning forward to give a large horizontal component to any force on impact. This is most easily obtained if the tackler is moving so that the driving force from the ground contact is forward to counteract the torque of the weight when the centre of mass is ahead of the feet.
During impact the ground contact will effectively increase the mass and hence momentum of the tackler. If the tackler’s body remained completely rigid, the carrier would be stopped dead or driven back at the speed of the tackler. This is of course not possible if the tackler is moving or has flexed joints, but is approached if the tackler ensures that ground contact is maintained in the tackle and that the legs drive through the tackle. In this case the tackler may take shorter strides before and during contact to maintain the driving force. A forward lean will also cause the tackler to pivot about the tackler’s feet as the angular momentum of the carrier about the tackler is transferred to the combined system, carrying the carrier upward from the ground. At this point the tackler is in ground contact and can rotate the carrier so that the tackler lands on top and the carrier is on his back putting the tackler in control. Continued drive by the tackler in a head on tackle can stop the carrier and drive them backward without rotation to land on their back, tackler on top.
In any tackle other than from behind, the tackler aims to ensure that their head is behind the carrier or at least on the side from which the tackler is approaching so that the head is not struck in the tackle. To achieve this, the tackler attempts to stay on the side of the tackler on which the contact will be made so that any change in direction can be allowed for. It will also tend to put the tackler on top after it is complete.

A substantial proportion of tackles in a game are not clean one-on-one challenges but are low speed upright mauls in which neither party is dominant.

Wednesday, May 10, 2006

Sidestep and swerve

Change in direction.
The track athlete either runs straight or around a fixed curve of relatively large radius (indoor sprints on a curved track are now rare). The rugby player frequently changes direction with a large radius curve when outflanking the defence using pace, medium radius in a swerve to beat a player or with a very small radius in a sidestep to wrong foot a defender.
To achieve any change of direction it is necessary to apply a force that is not parallel to the motion of the runner. If the change in direction is to be achieved without change in speed the net force will be at right angles to the direction of motion and acts toward the inside of the curve or toward the centre of curvature. This type of force is called a centripetal force (centri – centre, petal – seeking).
This force produces an acceleration with no change in speed since an acceleration is the rate of change of velocity and velocity is a vector quantity which includes both speed and direction.
To find the acceleration and thus calculate the force we calculate the vector change in velocity:
using vector diagrams or using the formula for circular motion at a constant speed: mv^2/r
Note that the faster (greater v) the runner goes the much greater the force needed to change direction and the sharper the direction change (smaller r) the greater the force needed.
For example, a runner in the inside lane of the track must run a bend of radius about 35m and is sprinting at about 10m/s at the apex of the curve. For an 80kg runner, therefore, the force required across the direction of motion is 228N.
In accelerating at the start of the race a sprinter the average rate is about 4.0ms-2 indicating a force of about 320N. Taking this as the force that an athlete can exert through ground contact while running, the net force available for running tangentially is about225N so that the maximum speed that can be maintained on the bend is substantially less than that at the same stage of a straight run.
If the athlete is in the outside lane, however, the radius of the bend is increased to about 40m. The force now needed is 200N and the available tangential component is 250N or approximately 10% more than in the inside lane.
The centripetal force is provided by a transverse reaction or frictional force from the track. To provide this force while continuing to support the runner’s body, the leg must attack the ground at a sufficient angle. As the angle from the vertical increases to increase the transverse force, the total force increases disproportionately. E.g. doubling the centripetal force at a low level might increase the total force by 12% while a further doubling raise the total by a further 40%. Continuing to increase the force in this way will eventually exceed the ability of the leg to cope while continuing to run. (nature jan 2006 greyhounds). This discussion has been based on average forces but the leg is not in continuous ground contact and is thus giving a more impulsive change in momentum resulting in much greater peak forces and thus likelihood of damage. To limit this, the runner increases the time of contact of the foot to reduce the force for the same impulse. The recovery phase is not altered as the limit to speed is the rate at which leg recovery occurs and is at its shortest at top speed. The increased time for each stride without change in stride length means that the cadence is decreased and the athlete slows.
In the photograph of young athletes running a 200m race the lean is measured as about 10deg, with the angle decreasing from inside lane to outside lane.
Taking the weight of a young athlete as about 600N we get a transverse component of force from Wtanq= F
or 600tan10 = 106N
Taking the speed of these athletes as 8m/s the transverse (centripetal) force should be: 96N
This is in excellent agreement with the calculated value.
This analysis assumes that the centripetal force is continuous, but in reality it must be a series if impulses which occur only when a foot is in contact with the track. During a straight sprint this can be as little as 40% of the time, but in a driving phase, as in accelerating around a bend, it may be nearer to 70% of the time. Accordingly when the foot is in contact with the ground it will exert a greater force than the average centripetal force calculated above. The calculated transverse force from the angle of lean is indeed larger than the centripetal force.
Attempting to increase the lateral force other than by lean will be unsuccessful as the transverse reaction force will also exert a torque about the centre of mass of the athlete tending to make the upper body rotate outwards. To balance this, the athlete now leans towards the centre of the bend so that the normal or vertical reaction force now exerts a counterbalancing torque about the centre of mass. The greater the centripetal force the more the athlete will have to lean to give the required moment arm for his weight.
Transferring these principles to rugby we see that the looping run which increases the distance from the defence has such a large radius of curvature that the centripetal force has little limiting effect on the speed at which the player is running and allows the fast player who has sufficient room to move to out pace a slower opposition by making them run farther to the tackle position.
The swerve is usually carried out with a small radius curve requiring a substantial inward lean with the upper body much closer to the opposition than the lower body. This will make it difficult to maintain a high speed since the driving phase will have only a limited forward component during the manoeuvre. The running action when several strides are taken during the swerve is not as efficient as straight line motion. The lean does, however make it difficult to tackle the ball carrier low and leaves the tackler vulnerable to a fend if the attempt is higher. The fend will of course give the attacker additional support and enable the swerve to be completed more easily.
The sidestep is designed to carry the ball carrier away from the defender suddenly so that the tackler cannot adjust before the attacker is out of reach. It is normally achieved by a single very forceful thrust of the outside leg which gives a large impulse to change the direction sharply. To maximise the impulse the runner maintains contact for as long as possible preventing excess strain on the propping leg. Since the leg is often placed forward and to the outside to give the required impulse, the speed of the ball carrier is often reduced substantially and the side step must be followed immediately by forceful acceleration in the forward direction. It is noticeable that players who attempt to sidestep repeatedly in one run end up almost stationary. For its efficacy, the sidestep relies on the inertia of the defender carrying him out of reach of the attacker or preventing him from matching the change in direction of the attacker. For this reason the attacker will tend to step back against the motion of the defensive line using the defender’s own motion to create the gap through which the attacker can now accelerate. In this situation the loss of momentum is less crucial since the attacker is accelerating from a position out of reach of the defender, who must first stop and reverse his motion before commencing pursuit. Defensive lines are set up to counter this change of direction by ensuring that the tackler approaches the attacker from the inside and that remaining defenders move across the field to prevent the attacker cutting back.
To create the situation where the defender is wrong footed by a sudden change in direction the attacker will perform an in-out manoeuvre in which he initially steps or swerves toward the centre of the field causing the defender to check or reverse his movement across the field, whereupon the attacker steps out again into the gap produced by the initial change in direction. when this is performed in close contact, the initial inward step is merely to give a stronger prop for the outward evasive step. Even if contact is not avoided it puts the attacker into the defender’s weaker zone some distance from the core.
The sidestep is a much more violent manoeuvre than curve running since the change of direction which may be around 45deg occurs within a single stride.
Typically a runner travelling at 8m/s changes direction by45deg and slows to 6m/s in executing a sidestep. Video of the driving phase shows that the propping foot maintains contact with the ground for about 0.24s (6 frames at 25 frames/s). Finding the vector change in velocity and dividing by the time gives an acceleration of about 20m/s/s at 132deg to the original line of travel. That is to say, a force of approximately twice the weight of the runner. Initial trials on a three axis force plate by an amateur player showed peak forces of twice the weight and average forces exceeded the weight by 50% while in a simple run through the peak force equalling the weight and averaging less than half the weight. The time of contact in the sidestep was approximately 0.25s while the running step lasted only 0.15s, the impulse needed to maintain running being much less than that required to perform the sidestep.

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.