A Review of Vectors Vector Mechanics in the Game of Pool Richard Damian Dimitry Hikin Wai Feng Xu SC441L New York City Technical College May 17, 2002

Overview Review of Experiment Review of Experiment Vector Terminology Vector Terminology Vector Addition Vector Addition Data and Analysis Data and Analysis Summary of Results Summary of Results Vectors in the Game of Pool Vectors in the Game of Pool Introduction to Pool Introduction to Pool Application of Vectors in Pool Application of Vectors in Pool “Ride the Nine” Shot Analysis “Ride the Nine” Shot Analysis Conclusion Conclusion

Vector Terminology vector n. A quantity completely specified by a magnitude and a direction. vector n. A quantity completely specified by a magnitude and a direction. Velocity Velocity Displacement Displacement resultant n. A single vector that is the equivalent of a set of vectors. resultant n. A single vector that is the equivalent of a set of vectors. equilibrant n. A vector equal and opposite to the resultant. equilibrant n. A vector equal and opposite to the resultant. component n. One of a set of two or more vectors having a sum equal to a given vector. component n. One of a set of two or more vectors having a sum equal to a given vector.

Vector Addition Graphical Method Graphical Method [first approach] Graphical Method [first approach] Place tail of vector B at the head of vector A Place tail of vector B at the head of vector A Resultant = vector from tail of A to head of B Resultant = vector from tail of A to head of B B A R A B R E

Vector Addition Graphical Method (cont’d) Graphical Method [second approach] Graphical Method [second approach] Place tail of A 1 with tail of B 1 Place tail of A 1 with tail of B 1 Form a parallelogram Form a parallelogram Duplicate vectors A 1 and B 1 to create A 2 and B 2 Duplicate vectors A 1 and B 1 to create A 2 and B 2 Place tail of A 2 at head of B 1, tail of B 2 at head of A 1 Place tail of A 2 at head of B 1, tail of B 2 at head of A 1 Resultant = vector from tail of A 1 and B 1 to head of A 2 and B 2 Resultant = vector from tail of A 1 and B 1 to head of A 2 and B 2 B2B2 A1A1 R A2A2 B1B1 A1A1 B1B1 R E

Vector Addition Component Method Component Method Component Method x axis is labeled i x axis is labeled i y axis is labeled j y axis is labeled j Get the x and y values of A and B A x I, A y j, B x I, B y j, A A y j Ax iAx i B B y j Bx iBx i A y j B y j Ax iAx iBx iBx i Add x and y components R E

Data and Analysis Trial 1

Data and Analysis Trial 2

Data and Analysis Trial 3

Summary of Results Graphical Method Percent Error Graphical Method Percent Error 1 st Trial – 0.77% 1 st Trial – 0.77% 2 nd Trial – 0.38% 2 nd Trial – 0.38% 3 rd Trial – 2.27% 3 rd Trial – 2.27% Component Method Percent Error Component Method Percent Error 1 st Trial – 0.38% 1 st Trial – 0.38% 2 nd Trial – 6.03% 2 nd Trial – 6.03% 3 rd Trial – 1.95% 3 rd Trial – 1.95% Sources of Error Sources of Error Unbalanced force table Unbalanced force table Strings touching edge of table – friction Strings touching edge of table – friction Ring in center of table was not completely centered Ring in center of table was not completely centered Human error when setting balance Human error when setting balance

Introduction to Pool Pool is Physics Pool is Physics The Pool Table The Pool Table

Introduction to Pool (cont’d) Pool Balls Pool Balls Weight: 6oz or 170gm Weight: 6oz or 170gm Diameter: 2.14in or 5.7cm Diameter: 2.14in or 5.7cm Classification in Game Classification in Game Numerical – 1-15 Numerical – 1-15 Solids & Stripes Solids & Stripes 1-7 Solids 1-7 Solids 9-15 Stripes 9-15 Stripes Standard Balls Elephant Beautiful Balls

Application of Vectors in Pool Cue-ball aim Cue-ball aim Cue-ball and object-ball direction Cue-ball and object-ball direction Ball direction after hitting rail Ball direction after hitting rail Cue-ball placement Cue-ball placement Object-ball positioning Object-ball positioning

“Ride 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 (cont’d) Reverse angle of previous photo

“Ride the Nine” Analysis 1 st Video – “Failed Shot.mpg” 1 st Video – “Failed Shot.mpg” Cue ball hit directly into 1-ball Cue ball hit directly into 1-ball No angle change No angle change Resultant has same direction as cue and object ball Resultant has same direction as cue and object ball 2 nd Video – “Successful Shot.mpg” 2 nd Video – “Successful Shot.mpg” Altered direction of cue ball to slice 1-ball into side pocket Altered direction of cue ball to slice 1-ball into side pocket Cue ball travels down the rail to sink 9-ball into corner pocket Cue ball travels down the rail to sink 9-ball into corner pocket Analysis of Successful Shot Analysis of Successful Shot Vectors of cue-ball (after collision) = A Vectors of cue-ball (after collision) = A Vector of 1-ball (after collision) = B Vector of 1-ball (after collision) = B A + B = R A + B = R Equilibrant is path that cue ball originated from Equilibrant is path that cue ball originated from

Conclusion Vectors have both magnitude and direction Vectors have both magnitude and direction Vectors can be added geometrically Vectors can be added geometrically Add x and y values of vectors to find resultant Add x and y values of vectors to find resultant Pool is a vector game Pool is a vector game Vectors help players aim for shots Vectors help players aim for shots Magnitude and direction of ball movement determines stopping position Magnitude and direction of ball movement determines stopping position