TP A.18 Distance required for stun and natural roll to develop for different tip offsets

Transcription

1 technical proof technical proof TP A.18 Distance required for stun and natural roll to develop for different tip offsets supporting: The Illustrated Principles of Pool and Billiards y David G. Alciatore, PhD, PE ("Dr. Dave") originally posted: 8/9/5 last revision: /1/9 all size and physical parameters: D.5in g sec μ. 9 max 16 in D gravity typical sliding friction etween all and tale cloth maximum suggested tip offset (see TP A.1) typical all speeds: v slow 3mph v medium 7mph v fast 1 mph From TP A.1, the forward angular spin () is related to tip offset aove center () and all speed (v) according to: ω( v) 5 v ( 1) From TP 4.1, the time required for stun to develop (with <) is: t s = ω α = ( ω) 5μg Sustituting Equation 1 gives: t s = 5( ) v 5μg = v μg ( )

2 From TP 4.1, the time required for roll to develop is: t d = sign( v ω) ( v ω) 7μg Sustituting Equation 1 gives: 5 v 5 t d = sign1 1 ( 3) 7μg Distance traveled is related to time according to: d = vt 1 μg t ( 4) Sustituting Equation into Equation 4 gives the distance required for stun to develop: = v ( v ) v μg 1 μg v μg v 1 1 μg ( 5) Sustituting Equation 3 into Equation 4 gives the total distance for sliding to stop and rolling to egin: d v 5 sign 1 v μg μg v 5 = 1 7μg dv ( ) sign 1 5 v 1 7μg 5 v μg 5 4 ( 6) With a non-overspin shot, Equation 6 ecomes: v d = 98μg d = v μg This equation can e verified at special cases: 5 gives d = which agrees with TP 4. = gives 1v d = 49μg which agrees with TP 4.1

3 Here are typical values for different types of shots: stun-drag shot: in.737 dv slow 4.1 dv medium dv fast draw-drag shots: half of the maximum recommended tip offset:.658 v slow v medium max v fast dv slow 5.84 dv medium dv fast maximum recommended tip offset: max v slow 1.31 dv slow v medium dv medium 18.5 v fast.999 dv fast follow shots: half of the maximum recommended tip offset: dv slow sign v slow ωv slow dv medium maximum recommended tip offset: sign v slow ωv slow.7 dv slow dv medium max.5 at this tip offset, the vertical spin is less than roll to egin with max dv fast.5 at this tip offset, the vertical spin is more than roll to egin with dv fast

4 Below, we will look at how the distance for stun to develop ( ) compares to the entire distance for sliding to stop and rolling to egin (d). We will also look at how these distances vary for drag shots of different cue all speeds (v) and tip offsets (< implies a elow-center hit). (Note - see TP A.3 to see how cue all speed varies with cue speed and tip offset.) small 5% max medium 5% max large 75% max max max 5 5 max v v slow v slow.1mph v fast v slow v medium v fast max % The distance to stun increases fairly linearly with tip offset, and is more sensitive to tip offset changes at higher speeds: medium v fast medium medium v medium medium v fast v medium.939 Douling the offset approximately doules the stun distance, at all speeds: medium v medium medium v medium. medium v fast medium v fast. Douling the speed increases the stun distance y a factor of 4, at all speeds: max v medium max v medium 4 max v fast max v fast 4

5 15 small v medium v large v v mph The distance to stun is more sensitive to speed changes at larger offsets: large v medium large medium v medium medium v medium v medium Douling the speed increases the stun distance y a factor of 4, at all offsets: medium v medium medium v medium 4 large v medium large v medium 4 Douling the speed increases the stun distance y a factor of 4, at all offsets: large v medium large v medium 4 medium v medium medium v medium 4

6 dv slow dv medium dv fast max % The distance to roll increases fairly linearly with tip offset, and is more sensitive to tip offset changes at higher speeds. mall v d medium v d large v v mph The sensitivity of roll distance to speed change is only a little greater for larger offset hits.

7 Here's how the ratio of stun distance to roll distance varies with tip offset (it doesn't vary with speed): d medium v fast medium v fast 1.63 d medium v medium medium v medium 1.63 v v medium dv ( ) ( v ) max % 5% max dv ( ) ( v ) 75% max.6 5% max dv ( ) ( v ) dv ( ) ( v ) % max dv ( ) ( v ) dv ( ) ( v) ( v ) % For large offset drag shots, the distance for roll to develop aer stun is a small fraction (aout 15%) of the distance required for stun to develop.