TP A.29 Using throw to limit cue ball motion

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1 techical proof TP A.9 Usig to limit cue ball motio supportig: The Illustrate Priciples of Pool a Billiars by Dai G. Alciatore, PhD, PE ("Dr. Dae") techical proof origially poste: 5/18/7 last reisio: 6/11/15 The illustratio below efies the pertiet termiology a shows how ( ) is relate to agle ( ) a cut agle (). The cue ball has iitial elocity a fial elocity, a the object ball has fial elocity. The cue ball is assume to hae pure siespi () at cotact (i.e., stu ). A goal is to limit the cue ball motio to the right (x irectio) while achieig the esire agle to the left. - - R t t r - y x (a) a cut i same irectios (b) a cut i opposite irectios

2 From TP A.5, the cue balls fial spee compoet i the ormal () irectio is: where e is the coefficiet of restitutio. (1 e)cos ( 1) Also, the chage i spee i the ormal irectio for both balls is relate to the ormal impulse with: 1 ecos m For gie ball coitios, the cut agle a amout of Eglish create a certai amout of (see TP A.14). The agle relates the tagetial a ormal spees a impulses, a ca be expresse (from TP A.5) as: ta t t 3 Therefore, usig Equatio, the chage tagetial spee of both balls ca be expresse as: t m m 1 ecos t ta ta 4 Therefore, the cue balls fial spee compoet i the tagetial irectio (see TP A.5) is: t t 1 si e m si 1 cos ta 5 Whe the cut agle () is positie (i.e., i the same irectio as the, as show i figure "a" aboe), both the cut agle a terms a to the tagetial elocity. Whe the cut agle is egatie (i.e., i the opposite irectio from the, as show i figure "b" aboe), the cut-agle term subtracts from the portio of the tagetial elocity. For a cue-ball "hol" or "kill", the goal is to hae the cue ball moe as little as possible to the right (i the x irectio). A goo measure for how much the cue ball moes to the right after cotact is the x-compoet of the elocity after cotact. I will refer to this as the cue-ball "rift spee." It ca be fou from the -t compoets, which are at agle relatie to the x-y axes: x si cos 6 t Likewise, the object ball elocitys x-compoet ca be fou with: x si cos 7 t

3 For the remaier of the aalysis, I will assume a perfectly elastic collisio (i.e., e=1, which ist far from the typical alue of about.94). With this, the fial cue ball rift spee ca be expresse from Equatios 1, 5, a 6 as: x si cos ta cos 8 To compare the cue-ball hol-effectieess of oe s. aother, assumig the esire agle ( ) is the same, it is useful to express the cue balls fial x-irectio spee as a percetage of the fial object ball spee. The fial object ball spee ca be expresse as: ( 9) t where the object balls fial -t elocity compoets are gie by Equatios a 4. Agai, for e=1, this ca be expresse as: 1 ta cos 1 Therefore, the hol effectieess ca ow be measure with the followig rift spee, expresse as a percetage of the fial object ball spee: rift x cos ta ta 1 ta ( 11) Similarly, usig Equatios, 4, 7, a 11, the object-ball rift percetage ca be expresse as: si cos ta 1 ta x rift ( 1) For a esire total agle, a for a gie amout of, the require cut agle is (see figure "a" aboe): where is: 13 ( 14) Agle ca be fou from the triagle show to the left of figure "a" aboe. The followig two equatios relate the ertical a horizotal istaces i the triagle: ( 15) R cos r cos( ) Rsi r si ( 16)

4 Solig Equatios 15 a 16 for, usig Equatio 14 gies: si ta 1 cos( R ) ( 17) So, from Equatios 13, 14, a 17, the cut agle is relate to the a agles with: si ta 1 18 cos( ) R To see if ca be use to help kill cue ball motio for a with a require agle, we ca compare the cue balls rift spee with to the rift spee without (e.g., if gearig outsie Eglish is use, or if CIT is eglecte). With o ( =), usig Equatios 11, 14, a 18, the rift-spee percetage ca be expresse as: rift o _ cos ta ( 19) With, the rift-spee percetage ca be expresse as: riftwith _ cos ta ta 1 ta ( ) The rift-spee percetage, with, ca actually be egatie, implyig that the cue ball actually rift i the same irectio as the (to the left). From the brackete term i the umerator, this occurs whe is less tha the egatie of the agle (see more below). Here is the MathCAD implemetatio of Equatios 17, 19, a : D.5i ata β θ θ D R R si θ θ cos θ θ ball iameter a raius DWT θ θ taθ θ β θ θ taθ cos θ θ 1 ta θ cosθ taθ β θ DNT θ

5 A typical alue for the maximum amout of possible, with a slow-spee stu- with the optimal amout of Eglish, is about 6 egrees. Usig this, we ca see how the rift-spee percetages ary with the agle. For this example, we will use a istace of 3 ball iameters. 3D θ 6eg θ eg.1eg 1eg 3 DWT θ θ DNT θ θ eg Regarless of the agle, the cue ball rift will be less if is use, as compare to ot usig (see the two cures aboe). For small agles, if the agle is large eough, the cue ball ca be stoppe completely a ee be mae to rift i the same irectio as the (i.e., the rift is egatie). For the example aboe, the maximum agle possible is: θ Gie eg = DWT θ θ θ Fi θ θ eg From Equatio 11, the limit where egatie cue ball rift is possible is always where the cut agle is the egatie of the agle, or from Equatio 18, where is the egatie of the agle : β θ θ eg

6 We ca also look at how the rift spee percetages ary with istace betwee the cue ball a object ball, for a gie agle a amout: θ 1eg D.1D 1D 5 DWT θ θ DNT θ D So whe the cue ball is closer to the object ball, it is much easier to use to reuce (a ee reerse) the cue ball rift. Howeer, ee for the small agle (1 eg) a large agle (6 eg) aboe, it is impossible to reerse or preet cue ball rift oce the balls are past a certai istace apart. For the example aboe, the istace is about 6 ball iameters (see the graph aboe or the calculatio below). 6D θ 1eg Gie = DWT θ θ Fi( ) D i

7 A iterestig challege is to try to create a agle (θ >) while stoppig the i place (DWT=). Below is a plot of the maximum agle possible for ifferet istaces () betwee the a, assumig a typical maximum alue of 6 egrees. θ 6eg max agle_with stop( ) rootdwtθ θ θ 1.1D1.D 1D 6 max agle_with stop( ) eg D The maximum agle possible approaches the maximum amout of possible as the gets closer to the ( = D). As the istace () betwee the a icreases, the maximum agle possible becomes less. Theoretically (eglectig swere), per the plot aboe, it is possible to hol the a create a agle at ay - istace. Howeer, at the table, there is a practical limit to the - istace oer which this ca be oe. The aalysis aboe has igore the effects of swere. Swere chages the effectie cut agle of the, makig it more ifficult to hol the. A at larger istaces, swere becomes more of a factor. Also, stu (for maximum ) is more ifficult to cotrol at larger istaces. Also, it is much more ifficult to juge squirt a swere a be accurate with the hit at larger istaces. Please refer to the "- hol" resource page i the FAQ sectio at billiars.colostate.eu for illustratios, examples, a ieo emostratios.

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Problem Set #3 Due Weesay, March 4, 009 Problem : Waa s worl a) The geometry or this problem is show i Figure. For part (a), the object