A Review of VectorsA Review of Vectors

Vector Mechanics in the Game of PoolVector Mechanics in the Game of Pool

Richard DamianDimitry HikinWai Feng Xu

SC441LNew York City Technical College

May 17, 2002

OverviewOverview Review of ExperimentReview of Experiment

Vector TerminologyVector Terminology Vector AdditionVector Addition Data and AnalysisData and Analysis Summary of ResultsSummary of Results

Vectors in the Game of PoolVectors in the Game of Pool Introduction to PoolIntroduction to Pool Application of Vectors in PoolApplication of Vectors in Pool ““Ride the Nine” Shot AnalysisRide the Nine” Shot Analysis

ConclusionConclusion

Vector TerminologyVector Terminology vector vector n. n. A quantity completely specified by a A quantity completely specified by a

magnitude and a direction.magnitude and a direction. VelocityVelocity DisplacementDisplacement

resultantresultant n. n. A single vector that is the A single vector that is the equivalent of a set of vectors.equivalent of a set of vectors.

equilibrant equilibrant n. n. A vector equal and opposite to A vector equal and opposite to the resultant.the resultant.

componentcomponent n. n. One of a set of two or more One of a set of two or more vectors having a sum equal to a given vectors having a sum equal to a given vector.vector.

Vector AdditionVector AdditionGraphical MethodGraphical Method

Graphical Method [first approach]Graphical Method [first approach]Place tail of vector B at the head of vector APlace tail of vector B at the head of vector AResultant = vector from tail of A to head of BResultant = vector from tail of A to head of B

B

A

R

A

B

R

E

Vector AdditionVector AdditionGraphical Method (cont’d)Graphical Method (cont’d)

Graphical Method [second approach]Graphical Method [second approach] Place tail of APlace tail of A11 with tail of B with tail of B11

Form a parallelogramForm a parallelogram Duplicate vectors ADuplicate vectors A11 and B and B1 1 to create Ato create A22 and B and B22

Place tail of APlace tail of A22 at head of B at head of B11, tail of B, tail of B2 2 at head of Aat head of A11

Resultant = vector from tail of AResultant = vector from tail of A11 and B and B11 to head of A to head of A22 and B and B22

B2

A1

R

A2

B1

A1

B1

R

E

Vector AdditionVector AdditionComponent MethodComponent Method

Component MethodComponent Method x axis is labeled x axis is labeled ii y axis is labeled y axis is labeled jj

Get the x and y values of A and B

Ax I, Ay j, Bx I, By j,

AAy j

Ax i

BBy j

Bx i

Ay j

By j

Ax i Bx i

Add x and y components

R

x

y

xx

yyy

xxx

x

y

y

xx

y

x

R

R

RRR

BAR

BAR

A

A

A

A

AAA

AA

AA

x

1

22

1

22

tan

tan

tan

sin

cos

E

Data and AnalysisData and AnalysisTrial 1Trial 1

Data and AnalysisData and AnalysisTrial 2Trial 2

Data and AnalysisData and AnalysisTrial 3Trial 3

Summary of ResultsSummary of Results

Graphical Method Graphical Method Percent ErrorPercent Error 11stst Trial – 0.77% Trial – 0.77% 22ndnd Trial – 0.38% Trial – 0.38% 33rdrd Trial – 2.27% Trial – 2.27%

Component Method Component Method Percent ErrorPercent Error 11stst Trial – 0.38% Trial – 0.38% 22ndnd Trial – 6.03% Trial – 6.03% 33rdrd Trial – 1.95% Trial – 1.95%

Sources of ErrorSources of Error Unbalanced force tableUnbalanced force table Strings touching edge Strings touching edge

of table – frictionof table – friction Ring in center of table Ring in center of table

was not completely was not completely centeredcentered

Human error when Human error when setting balancesetting balance

Introduction to PoolIntroduction to Pool

PoolPool isis PhysicsPhysics The Pool TableThe Pool Table

Introduction to Pool Introduction to Pool (cont’d)(cont’d)

Pool BallsPool Balls Weight: 6oz or 170gmWeight: 6oz or 170gm Diameter: 2.14in or Diameter: 2.14in or

5.7cm5.7cm Classification in GameClassification in Game

Numerical – 1-15Numerical – 1-15 Solids & StripesSolids & Stripes

1-7 Solids1-7 Solids 9-15 Stripes9-15 Stripes

Standard Balls

Elephant Beautiful

Balls

Application of Vectors in PoolApplication of Vectors in Pool

Cue-ball aimCue-ball aimCue-ball and object-ball directionCue-ball and object-ball directionBall direction after hitting railBall direction after hitting railCue-ball placementCue-ball placementObject-ball positioningObject-ball positioning

““Ride the Nine” ShotRide the Nine” Shot

Cue ball to be hit at 45 degrees towards 1-ball to sink in side pocket, continue to hit 9-ball to sink in corner pocket

““Ride the Nine” Shot Ride the Nine” Shot (cont’d)(cont’d)

Reverse angle of previous photo

““Ride the Nine” AnalysisRide the Nine” Analysis 11stst Video – “Failed Shot.mpg” Video – “Failed Shot.mpg”

Cue ball hit directly into 1-ballCue ball hit directly into 1-ball No angle changeNo angle change Resultant has same direction as cue and object ballResultant has same direction as cue and object ball

22ndnd Video – “Successful Shot.mpg” Video – “Successful Shot.mpg” Altered direction of cue ball to slice 1-ball into side pocketAltered direction of cue ball to slice 1-ball into side pocket Cue ball travels down the rail to sink 9-ball into corner pocketCue ball travels down the rail to sink 9-ball into corner pocket

Analysis of Successful ShotAnalysis of Successful Shot Vectors of cue-ball (after collision) = AVectors of cue-ball (after collision) = A Vector of 1-ball (after collision) = BVector of 1-ball (after collision) = B A + B = RA + B = R Equilibrant is path that cue ball originated fromEquilibrant is path that cue ball originated from

ConclusionConclusion

Vectors have both magnitude and directionVectors have both magnitude and direction Vectors can be added geometricallyVectors can be added geometrically Add Add xx and and yy values of vectors to find resultant values of vectors to find resultant

Pool is a vector gamePool is a vector game Vectors help players aim for shotsVectors help players aim for shots Magnitude and direction of ball movement Magnitude and direction of ball movement

determines stopping positiondetermines stopping position