TP A.30 The effect of cue tip offset, cue weight, and cue speed on cue ball speed and spin

Transcription

1 technical proof TP A.30 The effect of cue tip offet, cue weight, and cue peed on cue all peed and pin technical proof upporting: The Illutrated Principle of Pool and Billiard y Daid G. Alciatore, PhD, PE ("Dr. Dae") originally poted: /4/007 lat reiion: 0/7/04 efore ipact ' after ipact Linear oentu ut e conered during the ipact, o: ' If an elatic colliion i aued, energy i alo conered: ' The linear ipule etween the tip and all at ipact i equal to the change in oentu of oth the tick and all: F ' ˆ ( 3 ) and the angular ipule i equal to the change in angular oentu of the all: Fˆ ( 4)

2 Fro Equation, the final tick peed can e epreed a: ' and fro Equation 3 and 4, the pin rate of the all can e epreed a: Sutituting Equation and into Equation, and oling for the final all peed gie: 7 Oiouly, to achiee aiu all peed ( ) for a gien tick peed ( ), the offet () hould e 0 (i.e., center-all hit). For a typical a ratio (/3, per elow), the aiu all peed i related to the tick peed (neglecting energy loe) according to: So for a center-all hit, with typical equipent (and neglecting energy loe), the all oe 0% fater than the tick. With an etreely heay tick ( >> ), the theoretical liit for the aiu peed of the all i twice the peed of the tick. Sutituting Equation 7 into Equation gie the all pin rate: D N 9 For aiu all pin, for a gien cue-tick peed, the deriatie of with repect to ut e zero, o: 0 ' ' 0 ND DN d d 0 Soling thi equation for gie the tip offet that reult in the aiu all pin-rate:

3 Typical a alue are oz for the all and 8 oz for the cue tick. Thi gie a a ratio of /3, o the predicted offet for aiu pin i: ( ) When conidering the offet for aiu pin, we need to enure the all eparate fro the cue tick after ipact; otherwie, the tip would continue to puh and ru on the cue all after the initial ipact (if we neglect the effect of quirt and cue tick deflection and iration for the oent). The condition for eparation etween the tip and all i: ' ( 3) Equation and 7 can e ued to how that thi contraint i equialent to: ( 4) Coparing thi to Equation how that the offet that produce the aiu pin rate i alo on the erge of preenting cue tip and all eparation. So it would appear that 0.73 i till the optial alue for achieing aiu all pin. The analyi aoe ha aued the colliion i perfectly elatic (i.e., no energy i lot). Now, we'll look at the effect of an inelatic colliion etween the cue tip and cue all. In thi cae, the coefficient of retitution (e) i le than one, and energy i lot in the colliion. If repreent the efficiency of the colliion, Equation can e written a: ' Thi equation can e written (y ultiplying y / ) a: where r i the all-to-cue-tick a ratio: ' r r ( ) r 7 Sutituting Equation and into Equation and iplifying reult in: r 0 8 r

4 Equation 8 i a quadratic, and can e oled for the final all peed: r r ( 9) For an elatic colliion (e = ), no energy i lot (==00%), and Equation 9 reduce to Equation 7. To ee how the colliion efficiency () i related to the coefficient of retitution (e), let' look at the iple cae of a center-all hit (=0, =0). The coefficient of retitution (CO = e) i defined a the ratio of the peed of eparation and the peed of approach etween the cue tick and cue all: e peed of eparation peed of approach ' 0 Fro Equation and 0, it can e hown that final cue tick peed i and the final cue all peed i: re ' r e r Sutituting Equation and into Equation (with =0), gie: re r e r 3

5 Here' a plot of how the efficiency η arie with e and r, for typical range of alue: η e r r e r ( e) r e r_all r_ediu 9 r_large η e r_all η e r_ediu 0.88 η e r_large The efficiency and CO ary with cue tip type and the a of the cue tick. The CO (e) alo ight ary with tip offet and hot peed, ut I don't hae data to quantify thi. Fro HSV B.4, typical leather-tip playing-cue CO are in the 0.73 range. For a 9 oz cue, that correpond to an efficiency of: η e For the reainder of the analyi, I'll ue an efficiency of 0.87 (87%), correponding to an energy lo of 3%, which i fairly typical and i what Corioli ued in hi ook. η 0.87 We will now look at how all peed (fro Equation 9) and pin (fro Equation, epreed in unit of reolution per econd), ary with tip offet and cue a for a contant colliion efficiency:.in all radiu ph typical tick peed r η η r r rp r π r

6 rp r_all rp r_ediu rp r_large Therefore, with a typical colliion efficiency, the offet that produce the ot pin i cloe to the icue liit (0.). Note that, for the auption aoe, a lighter cue (with a larger a ratio r ), will delier lightly ore pin to the cue all than a heaier cue for offet cloe to the icue liit (/ = 0.), ut thi proaly in't enough of a reaon to try to witch to a lighter cue tick. The only other way to increae the aount of pin, other than y increaing the tip offet (up to the icue liit), i to increae the tick peed. It i intereting to copare the relatie effect of increaing tick peed. increaing tip offet. Let' look at increaing tick peed y 0%. increaing the offet y 0% (for 0% and 00% aiu offet, and at ediu peed): r 9 typical all-to-tick a ratio rp. r rp r rp. r rp r.. rp. r rp r rp. r rp r So increaing the tick peed ha the ae effect on the pin rate, regardle of the aount of offet. The pin rate increae y the ae aount a the peed. Howeer, increaing the tip offet ha a aller effect, and thi effect decreae at larger offet. Therefore, it i afer to keep the cue tip away fro the icue offet liit a little, and jut add ore tick peed to get ore pin. Thi i ore efficient than trying to add ore offet cloe to the liit, proided you can create ore tick peed without acrificing accuracy too uch.

7 Now let' look at how the cue tick and cue all peed after ipact ary with tip offet for different all-to-cue a ratio. Fro Equation, the cue tick peed after ipact ( ' ) i related to the cue all peed ( ) according to: r ' r Firt, let' look at a ediu-weight cue (8 oz): r r oz 8oz ' r ph r ph Therefore, cloe to the icue liit (/ = 0.) the cue all i at rik of not eparating fro the cue tip cleanly. Although, in reality, the cue tip deflect away fro the cue all after the hit, likely giing the cue all tie and ditance to clear.

8 Now let' look at a heaier cue ( oz): r oz oz 7 0. ' r ph r ph Therefore, with a heaier cue, the cue all will definitely not eparate cleanly fro the cue tip for offet greater than aout /=0.4, and doule hit, puhe, and/or icue would e likely.

9 Now let' look at how cue all peed arie with tick weight and colliion efficiency. eeer, fro Equation 3, colliion efficiency depend on oth the coefficient of retitution (CO or e) and the all-to-tick a ratio ( r ): η e r r e r ( e) r Fro Equation 9, for a center-all hit (=0), the cue all peed depend on the tick peed ( ), the all-to-tick a ratio ( r ), and the CO (e) according to: r e η e r η e r r r The following plot how how cue all peed (a a percentage of tick peed) arie oer a wide range of poile cue weight. A typical alue i ued for the CO (0.73), aed on HSV B.4 with a typical playing cue with a ediu-hardne leather tip. Thi correpond to an efficiency of aout 89% for a typical-weight cue (9 oz): oz e 0.73 r 9oz % η e r ph 4oz4.oz oz 40 oz e % oz Notice that for a gien cue peed, ore cue weight gie ore cue all peed, a one would epect. Although, the enefit of the added weight diinihe at at higher weight. Alo, in reality, it i ore difficult to troke a heaier cue at the ae peed a a lighter cue.

10 With a higher-efficiency colliion, the reult change a little. Fro HSV B.4, typical phenolic-tip reak cue hae a CO (e) a high a For a 9oz cue, that correpond to a colliion efficiency of aout 94%: e 0.87 r % η e r ph 4oz4.oz oz 0 oz e % With a perfectly elatic colliion (00% efficient), the theoretical liit for the all-peed-to-tick-peed percentage, for a ridiculouly-heay cue i 00%, a predicted y Equation 7. Thi i not realitic to achiee, ut one thing i clear: a higher-efficiency tip reult in ignificantly ore cue all energy. Here are typical alue coparing a leather-tip playing cue to a phenolic-tip reak cue: oz e leather 0.73 e phenolic 0.87 ph r 9oz Breaking power i related to the energy deliered to the cue all, which i related to the quare of the peed, o here i how the tip copare in ter of reaking power: oz.8 r e phenolic r e leather So a high-efficiency phenolic tip proide aout a 7% increae in reaking power oer a typical ediu-hardne leather tip, which i ignificant.

11 For a gien cue weight, cue all peed increae linearly with cue peed, a hown y the following plot (and a predicted y Equation 7): oz 9oz ph.ph ph r e r e ph ph If you could increae in your cue peed y 0%, for a gien weight cue, the cue all peed would alo increae y 0%, and the effectie increae in reaking power would e %:. =. If you were ale to generate the ae cue peed with a range of cue weight, you can alo increae reaking perforance y uing a heaier cue. Increaing cue weight fro 7 to oz (with a fied cue peed) would increae the cue all peed y.3%, which would correpond to an effectie increae in reaking power of 3%: oz oz ph r_light 7oz r_heay oz r_heay e r_light e.03.3 r_heay e r_light e So with a draatic increae in cue weight (7 oz to oz), the enefit i not a large a one ight epect, een if the heaier cue could e troked at the ae peed a the lighter cue (which i uually not the cae). So what reak cue weight hould you ue to get the ot power with the et tip? Oiouly, and a hown aoe, a heaier cue will delier ore peed to the cue all; howeer, it i ore difficult to generate tick peed with a heaier cue. The optial cue weight, allowing for the et coination of weight and peed will ary with each indiidual aed on ucle phyiology. For ore info, ee the cue weight FAQ page here: