Description of the game

The features of the game played by The Geordies in Division 3 of Thornbury & District Skittles League (TDSL) in South Gloucestershire vary from one alley to the next but in general are as follows:
Alley: About 25 feet long; 5 feet wide; ideally level and polished but usually exhibiting some vice, such as variation of camber or slope with tide in the River Severn.
Balls: Three in number, smooth and spherical (allegedly) when new, developing pits and chips with maturity, between 4¹/₂ and 5 inches in diameter, made of lignum vitae (self-lubricating) or composition (simulated wood).
Pins: Nine in number, about 10 inches high, between 4 and 5 inches in diameter, made of hardwood (traditionally beech or sycamore) in a variety of shapes. Our home alley's pins are Bristol style as shown here.
Layout of pins: 3x3 diamond although towards the end of a particularly draining match the nine pins may look more like a truncated tenpin formation (shown left) with inevitable difficulties for front first matches.
Objective: With each 'hand' of three balls to score as close to the maximum of 27 as possible. In practice, and in matches, the score tends to be somewhat lower, typically between 2 and 7 for each hand! So what's so difficult?

Analysis of TDSL league tables

Analysis by Ron Jenkins of data for the 2004/05 season demonstrated that league results are well described by Poisson statistics. The implication is that the outcome of each game is finely balanced and equivalent to the toss of a coin. Therefore, the team that can gain an edge consistently will stand a very good chance of topping the league.

In 2007/08 Ron was delighted to receive funding from the SRC (Skittles Research Council) to complete his investigation into coin tossing as a legitimate, and more efficient, method of deciding the outcome of a match. If adopted, the principal benefits would be a substantial reduction in stress and focussing of attention on the social dimensions of skittling. Once SRC clearance for publication has been received, the results and recommendations for implementing the 'Jenkins Rule' will be reported in The Geordies blog.

Collision cross-section for first ball

Since the collision cross-section for any individual pin is not significantly influenced by position and angle of delivery, there is no benefit to be gained from such considerations in front first games. There is also limited potential for influence in some other leagues where, for example, rules require players to have their leading foot in line with centreline of pins. However, in other circumstances, there is some advantage to be gained from a combination of position on the alley and angle of delivery which maximises the chance of hitting wood.

Effective pin diameter is governed by size of ball and size & shape of pins (illustrated here for the Bristol pin shape). Use of an effective diameter reduces the ball to an infinitesimal point. The length and width of the alley may also be expressed in units of effective pin diameter, with the origin of a co-ordinate system for the alley at mid point of the bowling line, with the y-axis oriented along the alley through the centreline of the pins.

The distance from bowler to front pin is about 25 feet and width of alley is about 5 feet so the best achievable angle to front pin is arctan 2.5/25, i.e. arctan 0.1 or about 5 degrees to centreline of pins.

This illustration approximates the situation at The Geordies home alley. It applies to someone like The Geordies Hon. Sec. who lobs balls in the general direction of the pins and is lucky to keep them out of the gutter. There is about 1 in 2 chance of hitting wood if the ball runs parallel to the centreline, increasing to about 4 in 5 for a right hander bowling from right hand side of alley.

Once pins fall, precision is crucial (unless all nine have fallen), and the chance of hitting any given pin for the Hon. Sec. on the Knot alley is about 1 in 10.

What about spin I hear you say. More of that later.

Stance, grip, swing & release

I don't yet know where I stand on this complex set of topics. As soon as I get into the swing of it and my analyses are complete I will post them to the site. Amongst the questions to be answered is whether it is best to adopt a tenpin style, running up to the baseline to release the ball? Or does a stationary stance minimise the margin for error introduced by a run up?

Skidding & rolling

For pure rolling, the velocity of that part of the ball which is in instantaneous contact with the alley must be zero. This requires that the velocity of the ball along the alley should be perfectly balanced by the rotational velocity of the ball's surface at the contact point. Any imbalance will give rise to skidding, in which case a frictional force will operate, the ball's linear speed will decrease and its rotation will increase until the velocity of the contact point is zero and pure rolling is occurring.

The speed of delivery is a major factor in both time from release and distance down the alley before pure rolling is established. It is therefore also a dominant inluence on the ability to swerve the ball.

Assuming that the ball is not rotating on release, the time taken (t) and distance travelled (s) before pure rolling are given by C.B. Daish (Chapter 14 of The Physics of Ball Games) as follows:

The coefficient of sliding friction (µ) for wood on wood is between 0.3 and 0.5 depending on the condition of the ball and alley surfaces, with the lower value appropriate to pristine balls and a newly laid alley and the higher value likely to apply to the conditions on our home alley. A value of 0.4 is used for the following indicative calculations. The gravitational acceleration (g) is relatively constant in the Thornbury area with a value of about 9.8 m/s² (although there is allegedly a significant mascon in the vicinity of Tytherington!) Observation of The Geordies suggests that speed of delivery (V) is typically in the range 1 to 15 m/s.

Once pure rolling is achieved the linear velocity will be about 0.7V. The coefficient of rolling friction is between 0.002 and 0.05 and will not slow the ball significantly before pins are hit (or not). For values of V less than 11 m/s, about half of the initial kinetic energy will have been converted into rotation by the time that the pins are reached.

Swerve

There are three components of spin, only one of which directly affects a skittler's ability to swerve the ball:

• About a vertical axis - will not affect the ball's trajectory along the alley but may come into play on collision with pins.
• About a horizontal axis across the alley - this affects skidding but apart from the effect on speed down the alley has no direct impact on swerve.
• About a horizontal axis along the alley - this is the component that creates swerve. Swerve occurs throughout the ball's motion down the alley. The greater the speed of delivery, the shorter the length of time over which swerve can occur. Spin about a horizontal axis normal to the alley is virtually instantaneously converted into pure rolling. A ball projected at 1 m/s with spin time 1 s about that axis will move about 0.15 m (6 in) across the alley by the time it reaches the pins. Either increasing the speed of projection by a factor of 3 or reducing the spin by a factor of 3 will result in a movement across the alley of only 0.05 m. So the secret for maximum swerve is to deliver the ball at low speed while generating spin about this axis, aiming for a trajectory that will be at least 5 degrees to the alley centreline when the ball hits the front pin.

Bounce

Speed

[It may be worth emphasising here that the only recreational drug used by The Geordies is paracetamol which assists morning-after recovery following a hard night's skittling.]

The speed of delivery influences the following:

• Momentum available to share with pins - assuming a collision. Momentum is proportional to speed. For initial speeds less than 10 m/s, linear momentum will have been reduced to about 70% of its initial value when the pins are reached. For speeds above 10 m/s, less of the initial energy will have been converted into rotation and a greater proportion of linear momentun will be retained.
• Ability to overcome alley condition - the higher the speed of release, the lesser deflection caused by variations in alley level, imperfections, etc.
• Ability to swerve the ball - see above.