 ### Slide-roll transition

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 and initial spin are major factors 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:

T = 2V / 7µg ; S = 12V2 / 49µg

where V is the speed of delivery (m/s),
µ is the coefficient of sliding friction,
and g is gravitational acceleration (m/s2).

The coefficient of sliding friction (µ) for wood on wood is between 0.2 and 0.5 depending on the conditions of the ball and alley surfaces. The lower value is appropriate to pristine balls on a newly laid and polished alley. A value towards the upper end of the range is more likely to apply to the conditions on our home alley. The gravitational acceleration (g) is relatively constant in the Thornbury area with a value of about 9.81 m/s2 (although there is allegedly a significant mascon in the vicinity of Tytherington!) Observation of Geordies suggests that speed of delivery (V) is typically in the range 1 to 5 m/s perhaps as high as 10 m/s, occasionally 15 m/s, for Farmer Hall.

The following figure illustrates behaviour for a friction coefficient of 0.3. A ball launched at 5 m/s will slide for 2 m and will roll thereafter. A ball launched at 10 m/s under these conditions will be sliding for 8 m, i.e. most of the length of the alley. 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).

In preparation for studying swerve it is convenient to write all equations in terms of dimensionless distances and times normalised to the point of transition from skidding to pure rolling.

The velocity (v) of the ball and its distance travelled as a function of time are given by:

v/V = 1 - 2t/7T when t<T; v/V = 5/7 when t>T
s/S = 7/6 (t/T) - 1/6 (t/T)² when t<T; s/S = 1/6 + 5/6 (t/T) when t>T