The Physics of Bowling: How Good Bowlers Stay Off the Straight And

Physics of Bowling

Brody Dylan Johnson The Physics of Bowling:

Introduction How good bowlers stay oﬀ the straight and narrow.

Bowling Strikes Underlying Brody Dylan Johnson Physics

Mathematical Model Saint Louis University Human Experiment

Brody Dylan Johnson Physics of Bowling Outline

Physics of Bowling

Brody Dylan Johnson 1 Introduction

Introduction

Bowling Strikes 2 Bowling Strikes

Underlying Physics 3 Underlying Physics Mathematical Model

Human Experiment 4 Mathematical Model

5 Human Experiment

Brody Dylan Johnson Physics of Bowling The origin of ten pin bowling is attributed to an 1841 law in Connecticut banning ninepin bowling, which was the form of bowling brought to America from Europe. (The extra pin prevented violation of the law.) Lane dimensions: Length: 60 feet, Width: 41.5 inches. Ball speciﬁcations: Circumference: 2.25 feet, Weight: up to 16 pounds.

Brief History of Bowling1

Physics of Bowling

Brody Dylan Johnson Forms of bowling can be traced back to Egypt around 3200 B.C., although the ﬁrst “standardized rules” were Introduction established in New York in 1895. Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

1Reference: http://en.wikipedia.org/wiki/Bowling Brody Dylan Johnson Physics of Bowling Lane dimensions: Length: 60 feet, Width: 41.5 inches. Ball speciﬁcations: Circumference: 2.25 feet, Weight: up to 16 pounds.

Brief History of Bowling1

Physics of Bowling

Brody Dylan Johnson Forms of bowling can be traced back to Egypt around 3200 B.C., although the ﬁrst “standardized rules” were Introduction established in New York in 1895. Bowling Strikes The origin of ten pin bowling is attributed to an 1841 law Underlying Physics in Connecticut banning ninepin bowling, which was the

Mathematical form of bowling brought to America from Europe. (The Model extra pin prevented violation of the law.) Human Experiment

1Reference: http://en.wikipedia.org/wiki/Bowling Brody Dylan Johnson Physics of Bowling Ball speciﬁcations: Circumference: 2.25 feet, Weight: up to 16 pounds.

Brief History of Bowling1

Physics of Bowling

Mathematical form of bowling brought to America from Europe. (The Model extra pin prevented violation of the law.) Human Experiment Lane dimensions: Length: 60 feet, Width: 41.5 inches.

1Reference: http://en.wikipedia.org/wiki/Bowling Brody Dylan Johnson Physics of Bowling Brief History of Bowling1

Physics of Bowling

Mathematical form of bowling brought to America from Europe. (The Model extra pin prevented violation of the law.) Human Experiment Lane dimensions: Length: 60 feet, Width: 41.5 inches. Ball speciﬁcations: Circumference: 2.25 feet, Weight: up to 16 pounds.

1Reference: http://en.wikipedia.org/wiki/Bowling Brody Dylan Johnson Physics of Bowling The base score for a frame is the total pinfall (number of pins knocked down); In the case of a strike (all 10 pins knocked down on ﬁrst ball) the pinfalls of the next two balls are added to the score for the frame; In the case of a spare (all remaining pins knocked down on second ball) the pinfall of the next ball is added to the score for the frame; These bonus pinfalls can lead to up to two additional frames when a spare or strike occurs in the tenth frame.

Scoring

Physics of Bowling

Brody Dylan There are ten frames in which up to two balls may be Johnson rolled toward the pins;

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling In the case of a strike (all 10 pins knocked down on ﬁrst ball) the pinfalls of the next two balls are added to the score for the frame; In the case of a spare (all remaining pins knocked down on second ball) the pinfall of the next ball is added to the score for the frame; These bonus pinfalls can lead to up to two additional frames when a spare or strike occurs in the tenth frame.

Scoring

Physics of Bowling

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling In the case of a spare (all remaining pins knocked down on second ball) the pinfall of the next ball is added to the score for the frame; These bonus pinfalls can lead to up to two additional frames when a spare or strike occurs in the tenth frame.

Scoring

Physics of Bowling

Brody Dylan Johnson Physics of Bowling These bonus pinfalls can lead to up to two additional frames when a spare or strike occurs in the tenth frame.

Scoring

Physics of Bowling

Brody Dylan There are ten frames in which up to two balls may be Johnson rolled toward the pins; Introduction The base score for a frame is the total pinfall (number of Bowling pins knocked down); Strikes Underlying In the case of a strike (all 10 pins knocked down on ﬁrst Physics ball) the pinfalls of the next two balls are added to the Mathematical Model score for the frame; Human Experiment In the case of a spare (all remaining pins knocked down on second ball) the pinfall of the next ball is added to the score for the frame;

Brody Dylan Johnson Physics of Bowling Scoring

Physics of Bowling

Brody Dylan Johnson Physics of Bowling 1 2 3 4 5 6 7 8 9 10 Score 9/1 10/- 10/- 10/- 10/- 9/1 9/1 9/1 9/1 9/1/9 224 9/1 10/- 9/1 10/- 9/1 10/- 9/1 10/- 9/1 9/1/9 198 These two games have identical frames with diﬀerent orderings. There are four strikes in each game.

The highest possible score without any strikes results from ten frames of 9-1 spares followed by an additional 9 in the eleventh frame, correponding to a pinfall of 19 each frame and a total score of 190. Consecutive strikes are essential to high scores.

Strategy

Physics of Bowling

Brody Dylan Johnson A perfect game consists of twelve consecutive strikes. The pinfall is 30 for each frame for a total score of 300. Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Consecutive strikes are essential to high scores.

Strategy

Physics of Bowling

Brody Dylan Johnson A perfect game consists of twelve consecutive strikes. The pinfall is 30 for each frame for a total score of 300. Introduction

Bowling The highest possible score without any strikes results from Strikes ten frames of 9-1 spares followed by an additional 9 in the Underlying Physics eleventh frame, correponding to a pinfall of 19 each frame Mathematical and a total score of 190. Model

Human Experiment

Strategy

Physics of Bowling

Brody Dylan Johnson A perfect game consists of twelve consecutive strikes. The pinfall is 30 for each frame for a total score of 300. Introduction

Human Consecutive strikes are essential to high scores. Experiment

Brody Dylan Johnson Physics of Bowling Strategy

Physics of Bowling

Brody Dylan Johnson A perfect game consists of twelve consecutive strikes. The pinfall is 30 for each frame for a total score of 300. Introduction

Human Consecutive strikes are essential to high scores. Experiment 1 2 3 4 5 6 7 8 9 10 Score 9/1 10/- 10/- 10/- 10/- 9/1 9/1 9/1 9/1 9/1/9 224 9/1 10/- 9/1 10/- 9/1 10/- 9/1 10/- 9/1 9/1/9 198 These two games have identical frames with diﬀerent orderings. There are four strikes in each game.

Brody Dylan Johnson Physics of Bowling The shot should make contact with the pocket, which is the space between Pin 1 and Pin 3. An angle of six degrees with respect to the lane boards is considered optimal for the generation of strikes2.

Angle of Attack

Physics of Bowling

Brody Dylan Johnson 7 8 9 10 Introduction

Bowling Strikes 4 5 6 Underlying Physics 2 3 Mathematical Model Human 1 Experiment

2Reference: http://www.jimloy.com/bowling/6degrees.htm Brody Dylan Johnson Physics of Bowling An angle of six degrees with respect to the lane boards is considered optimal for the generation of strikes2.

Angle of Attack

Physics of Bowling

2Reference: http://www.jimloy.com/bowling/6degrees.htm Brody Dylan Johnson Physics of Bowling Angle of Attack

Physics of Bowling

Brody Dylan Johnson 7 8 9 10 The shot should make Introduction contact with the pocket, Bowling which is the space Strikes 4 5 6 Underlying between Pin 1 and Pin 3. Physics 2 3 Mathematical An angle of six degrees Model with respect to the lane Human 1 Experiment boards is considered optimal for the θ generation of strikes2.

2Reference: http://www.jimloy.com/bowling/6degrees.htm Brody Dylan Johnson Physics of Bowling A six degree angle with a straight shot would require the bowler to stand about 6.3 feet to the side of the pocket, which is more than halfway across the adjacent lane. In order to achieve the desired six degree angle, the ball must be curved. A curved path requires the ball to experience acceleration. What forces can produce such an acceleration?

Reality Check

Physics of Bowling Assuming that the pocket is in the middle of the lane, the Brody Dylan Johnson largest angle possible with a straight ball is

Introduction −1 1.75 Bowling tan ≈ 1.67 degrees. Strikes 60

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling In order to achieve the desired six degree angle, the ball must be curved. A curved path requires the ball to experience acceleration. What forces can produce such an acceleration?

Reality Check

Physics of Bowling Assuming that the pocket is in the middle of the lane, the Brody Dylan Johnson largest angle possible with a straight ball is

Introduction −1 1.75 Bowling tan ≈ 1.67 degrees. Strikes 60

Underlying Physics A six degree angle with a straight shot would require the Mathematical bowler to stand about 6.3 feet to the side of the pocket, Model

Human which is more than halfway across the adjacent lane. Experiment

Brody Dylan Johnson Physics of Bowling A curved path requires the ball to experience acceleration. What forces can produce such an acceleration?

Reality Check

Physics of Bowling Assuming that the pocket is in the middle of the lane, the Brody Dylan Johnson largest angle possible with a straight ball is

Introduction −1 1.75 Bowling tan ≈ 1.67 degrees. Strikes 60

Underlying Physics A six degree angle with a straight shot would require the Mathematical bowler to stand about 6.3 feet to the side of the pocket, Model

Human which is more than halfway across the adjacent lane. Experiment In order to achieve the desired six degree angle, the ball must be curved.

Brody Dylan Johnson Physics of Bowling Reality Check

Physics of Bowling Assuming that the pocket is in the middle of the lane, the Brody Dylan Johnson largest angle possible with a straight ball is

Introduction −1 1.75 Bowling tan ≈ 1.67 degrees. Strikes 60

Underlying Physics A six degree angle with a straight shot would require the Mathematical bowler to stand about 6.3 feet to the side of the pocket, Model

Human which is more than halfway across the adjacent lane. Experiment In order to achieve the desired six degree angle, the ball must be curved. A curved path requires the ball to experience acceleration. What forces can produce such an acceleration?

Brody Dylan Johnson Physics of Bowling The velocity of the contact point B is given by:

vB = vC − ωC R

If vB = 0 then the ball is undergoing pure rolling. Otherwise the ball is sliding and frictional forces will act on the ball.

A Rolling/Sliding Ball

Physics of Bowling Suppose a ball (radius R) is rolling and sliding with Brody Dylan velocity v and angular rotation speed ω . Johnson C C ωC Introduction

Bowling Strikes C vC Underlying Physics

Mathematical B Model

Human Experiment

Brody Dylan Johnson Physics of Bowling If vB = 0 then the ball is undergoing pure rolling. Otherwise the ball is sliding and frictional forces will act on the ball.

A Rolling/Sliding Ball

Physics of Bowling Suppose a ball (radius R) is rolling and sliding with Brody Dylan velocity v and angular rotation speed ω . Johnson C C ωC Introduction

Bowling Strikes C vC Underlying Physics

Mathematical B Model

Human The velocity of the contact point B is given by: Experiment

vB = vC − ωC R

Brody Dylan Johnson Physics of Bowling A Rolling/Sliding Ball

Physics of Bowling Suppose a ball (radius R) is rolling and sliding with Brody Dylan velocity v and angular rotation speed ω . Johnson C C ωC Introduction

Bowling Strikes C vC Underlying Physics

Mathematical B Model

Human The velocity of the contact point B is given by: Experiment

vB = vC − ωC R

If vB = 0 then the ball is undergoing pure rolling. Otherwise the ball is sliding and frictional forces will act on the ball.

Brody Dylan Johnson Physics of Bowling If vB > 0 then ωC is too small for pure rolling and the friction force takes translational energy and converts it into rotational energy, increasing ωC and decreasing vC until vB = 0. If vB < 0 then ωC is too large for pure rolling and the friction force takes rotational energy and converts it into translational energy, increasing vC and decreasing ωC until vB = 0.

The Eﬀect of Friction

Physics of Bowling A frictional force FB will act on the ball opposite to the

Brody Dylan direction of vB . Johnson

Introduction C Bowling Strikes

Underlying FB vB Physics B

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling If vB < 0 then ωC is too large for pure rolling and the friction force takes rotational energy and converts it into translational energy, increasing vC and decreasing ωC until vB = 0.

The Eﬀect of Friction

Physics of Bowling A frictional force FB will act on the ball opposite to the

Brody Dylan direction of vB . Johnson

Introduction C Bowling Strikes

Underlying FB vB Physics B

Mathematical Model If vB > 0 then ωC is too small for pure rolling and the

Human friction force takes translational energy and converts it Experiment into rotational energy, increasing ωC and decreasing vC until vB = 0.

Brody Dylan Johnson Physics of Bowling The Eﬀect of Friction

Physics of Bowling A frictional force FB will act on the ball opposite to the

Brody Dylan direction of vB . Johnson

Introduction C Bowling Strikes

Underlying FB vB Physics B

Mathematical Model If vB > 0 then ωC is too small for pure rolling and the

Human friction force takes translational energy and converts it Experiment into rotational energy, increasing ωC and decreasing vC until vB = 0. If vB < 0 then ωC is too large for pure rolling and the friction force takes rotational energy and converts it into translational energy, increasing vC and decreasing ωC until vB = 0.

Brody Dylan Johnson Physics of Bowling Energy is destroyed when vC and ωC have opposite signs. Excess rotational energy can be thought of as a power source for the translational motion. The ball can accelerate as long as there is excess spin. Conversely, insuﬃcient rotational energy will result in a power drain on the translational motion and the ball will decelerate until a balance is reached.

Energy Considerations

Physics of Bowling

Brody Dylan Johnson Energy is conserved when v and ω have the same sign. Introduction C C

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Excess rotational energy can be thought of as a power source for the translational motion. The ball can accelerate as long as there is excess spin. Conversely, insuﬃcient rotational energy will result in a power drain on the translational motion and the ball will decelerate until a balance is reached.

Energy Considerations

Physics of Bowling

Brody Dylan Johnson Energy is conserved when v and ω have the same sign. Introduction C C Bowling Energy is destroyed when vC and ωC have opposite signs. Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Conversely, insuﬃcient rotational energy will result in a power drain on the translational motion and the ball will decelerate until a balance is reached.

Energy Considerations

Physics of Bowling

Brody Dylan Johnson Energy is conserved when v and ω have the same sign. Introduction C C Bowling Energy is destroyed when vC and ωC have opposite signs. Strikes

Underlying Excess rotational energy can be thought of as a power Physics source for the translational motion. The ball can Mathematical Model accelerate as long as there is excess spin.

Human Experiment

Brody Dylan Johnson Physics of Bowling Energy Considerations

Physics of Bowling

Brody Dylan Johnson Energy is conserved when v and ω have the same sign. Introduction C C Bowling Energy is destroyed when vC and ωC have opposite signs. Strikes

Underlying Excess rotational energy can be thought of as a power Physics source for the translational motion. The ball can Mathematical Model accelerate as long as there is excess spin. Human Conversely, insuﬃcient rotational energy will result in a Experiment power drain on the translational motion and the ball will decelerate until a balance is reached.

Brody Dylan Johnson Physics of Bowling Theoretically, if enough backspin were generated, the ball could actually reverse its direction of motion and come back towards the bowler. For the typical bowler, the ball is released with a good balance of rotational and translational energy and the ball will generally achieve a pure rolling state before reaching the pins.

Practical Observations

Physics of Bowling

Brody Dylan Johnson There are many bowlers that release the ball with a small amount of backspin. Generally the ball can be seen to Introduction reverse its direction of spin as the ball heads down the Bowling Strikes lane. Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling For the typical bowler, the ball is released with a good balance of rotational and translational energy and the ball will generally achieve a pure rolling state before reaching the pins.

Practical Observations

Physics of Bowling

Mathematical could actually reverse its direction of motion and come Model back towards the bowler. Human Experiment

Brody Dylan Johnson Physics of Bowling Practical Observations

Physics of Bowling

Mathematical could actually reverse its direction of motion and come Model back towards the bowler. Human Experiment For the typical bowler, the ball is released with a good balance of rotational and translational energy and the ball will generally achieve a pure rolling state before reaching the pins.

Brody Dylan Johnson Physics of Bowling Oil conditions: Although oil patterns vary from one place to the next, typically a bowling lane is oiled from the foul line to a point about ﬁfteen feet in front of the pins. This last section has a much higher coeﬃcient of friction due to the lack of oil and often contributes a noticeable redirection of the ball trajectory. (the “extra” spin must last long enough for this to work)

Mathematical Model, Part I

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling This last section has a much higher coeﬃcient of friction due to the lack of oil and often contributes a noticeable redirection of the ball trajectory. (the “extra” spin must last long enough for this to work)

Mathematical Model, Part I

Brody Dylan Johnson Physics of Bowling Mathematical Model, Part I

Physics of Bowling Physical parameters: Brody Dylan Johnson Mass = m Ball Radius = r Introduction 2 2 Bowling Rotational Inertia = I = 5 mr Strikes Coefficient of Friction = µ Underlying Physics Gravitational Constant = g Mathematical Oil conditions: Although oil patterns vary from one place Model to the next, typically a bowling lane is oiled from the foul Human Experiment line to a point about fifteen feet in front of the pins. This last section has a much higher coefficient of friction due to the lack of oil and often contributes a noticeable redirection of the ball trajectory. (the “extra” spin must last long enough for this to work)

Brody Dylan Johnson Physics of Bowling System of differential equations: One obtains a linear system of first order differential equations by applying Newton’s 2nd law to the bowling ball. (Both translational and rotational forms.)

Mathematical Model, Part II

Physics of Bowling

Brody Dylan Variables: We will use a vector Johnson x = x1 x2 x3 x4 x5 x6 Introduction x1 = position in lane (from left) Bowling Strikes x2 = position along lane (from foul line)

Brody Dylan Johnson Physics of Bowling Mathematical Model, Part II

Physics of Bowling

Brody Dylan Variables: We will use a vector Johnson x = x1 x2 x3 x4 x5 x6 Introduction x1 = position in lane (from left) Bowling Strikes x2 = position along lane (from foul line)

Underlying x3 = velocity in lane (left to right) Physics x4 = velocity along lane (towards pins) Mathematical Model x5 = rotation about x1 axis (backspin +) Human x6 = rotation about x2 axis (rightspin +) Experiment System of differential equations: One obtains a linear system of first order differential equations by applying Newton’s 2nd law to the bowling ball. (Both translational and rotational forms.)

Brody Dylan Johnson Physics of Bowling Friction Forces:

Fx1 = −µ m g sgn(vx1 ) Fx2 = −µ m g sgn(vx2 ) Newton’s second law:

m dx3/dt = Fx1 m dx4/dt = Fx2

Mathematical Model, Part III

Physics of Bowling

Brody Dylan (viewed from the right)(viewed from foul line) Johnson x6 x5

Introduction Bowling C x3 C x4 Strikes

Underlying Fx1 vx1 Fx2 vx2 Physics B B Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Newton’s second law:

m dx3/dt = Fx1 m dx4/dt = Fx2

Mathematical Model, Part III

Physics of Bowling

Brody Dylan (viewed from the right)(viewed from foul line) Johnson x6 x5

Introduction Bowling C x3 C x4 Strikes

Underlying Fx1 vx1 Fx2 vx2 Physics B B Mathematical Model

Human Friction Forces: Experiment

Fx1 = −µ m g sgn(vx1 ) Fx2 = −µ m g sgn(vx2 )

Brody Dylan Johnson Physics of Bowling Mathematical Model, Part III

Physics of Bowling

Brody Dylan (viewed from the right)(viewed from foul line) Johnson x6 x5

Introduction Bowling C x3 C x4 Strikes

Underlying Fx1 vx1 Fx2 vx2 Physics B B Mathematical Model

Human Friction Forces: Experiment

Fx1 = −µ m g sgn(vx1 ) Fx2 = −µ m g sgn(vx2 ) Newton’s second law:

m dx3/dt = Fx1 m dx4/dt = Fx2

Brody Dylan Johnson Physics of Bowling The complete system of diﬀerential equations: dx1 dx2 dt = x3 dt = x4 dx3 Fx1 dx4 Fx2 dt = m dt = m dx5 Fx2 r dx6 Fx1 r dt = I dt = − I . Notice that the components of acceleration for the ball are piecewise constant.

Mathematical Model, Part IV

Physics of Bowling

Brody Dylan Rotational form of Newton’s second law: Johnson

Introduction I dx6/dt = Fx1 r Bowling I dx /dt = −F r Strikes 5 x2

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Notice that the components of acceleration for the ball are piecewise constant.

Mathematical Model, Part IV

Physics of Bowling

Brody Dylan Rotational form of Newton’s second law: Johnson

Introduction I dx6/dt = Fx1 r Bowling I dx /dt = −F r Strikes 5 x2

Underlying Physics The complete system of diﬀerential equations: Mathematical dx1 dx2 Model dt = x3 dt = x4 Human dx3 Fx1 dx4 Fx2 Experiment dt = m dt = m dx5 Fx2 r dx6 Fx1 r dt = I dt = − I .

Brody Dylan Johnson Physics of Bowling Mathematical Model, Part IV

Physics of Bowling

Brody Dylan Rotational form of Newton’s second law: Johnson

Introduction I dx6/dt = Fx1 r Bowling I dx /dt = −F r Strikes 5 x2

Underlying Physics The complete system of diﬀerential equations: Mathematical dx1 dx2 Model dt = x3 dt = x4 Human dx3 Fx1 dx4 Fx2 Experiment dt = m dt = m dx5 Fx2 r dx6 Fx1 r dt = I dt = − I . Notice that the components of acceleration for the ball are piecewise constant.

Brody Dylan Johnson Physics of Bowling The ball experiences constant acceleration when it is sliding, which makes it possible for the ball to follow a parabolic path. The ball experiences no acceleration when it is in a pure rolling state and, thus, follows a linear path once it stops sliding. It is possible to achieve the six degree angle with a pure parabola: (f (x) = ax2 + bx + c, a = tan (π/30)/60, b = − tan (π/30), & c = 1.75; x in feet)

0 0 10 20 30 40 50 60 Parabolic trajectory

Implications of the Model

Physics of Bowling The ball does not curve faster with more spin, it just

Brody Dylan curves longer. (takes longer to reach pure rolling) Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling The ball experiences no acceleration when it is in a pure rolling state and, thus, follows a linear path once it stops sliding. It is possible to achieve the six degree angle with a pure parabola: (f (x) = ax2 + bx + c, a = tan (π/30)/60, b = − tan (π/30), & c = 1.75; x in feet)

0 0 10 20 30 40 50 60 Parabolic trajectory

Implications of the Model

Physics of Bowling The ball does not curve faster with more spin, it just

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling It is possible to achieve the six degree angle with a pure parabola: (f (x) = ax2 + bx + c, a = tan (π/30)/60, b = − tan (π/30), & c = 1.75; x in feet)

0 0 10 20 30 40 50 60 Parabolic trajectory

Implications of the Model

Physics of Bowling The ball does not curve faster with more spin, it just

Brody Dylan curves longer. (takes longer to reach pure rolling) Johnson The ball experiences constant acceleration when it is Introduction sliding, which makes it possible for the ball to follow a Bowling parabolic path. Strikes The ball experiences no acceleration when it is in a pure Underlying Physics rolling state and, thus, follows a linear path once it stops Mathematical sliding. Model

Human Experiment

Brody Dylan Johnson Physics of Bowling 3

0 0 10 20 30 40 50 60 Parabolic trajectory

Implications of the Model

Physics of Bowling The ball does not curve faster with more spin, it just

Human It is possible to achieve the six degree angle with a pure Experiment parabola: (f (x) = ax2 + bx + c, a = tan (π/30)/60, b = − tan (π/30), & c = 1.75; x in feet)

Brody Dylan Johnson Physics of Bowling Implications of the Model

Physics of Bowling The ball does not curve faster with more spin, it just

Human It is possible to achieve the six degree angle with a pure Experiment parabola: (f (x) = ax2 + bx + c, a = tan (π/30)/60, b = − tan (π/30), & c = 1.75; x in feet)

0 0 10 20 30 40 50 60 Brody Dylan JohnsonParabolicPhysics trajectory of Bowling 18 18 18

16 16 16

14 14 14

12 12 12

10 10 10

8 8 8

6 6 6

4 4 4

2 2 2

0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 1.60 degrees 5.38 degrees 3.75 degrees

Cheating the Geometry

Physics of Bowling The model allows us to compare trajectories obtained for

Brody Dylan various spins, speeds, and release angles. Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling 18 18

16 16

14 14

12 12

10 10

8 8

6 6

4 4

2 2

0 0 0 0.5 1 0 0.5 1 5.38 degrees 3.75 degrees

Cheating the Geometry

Physics of Bowling The model allows us to compare trajectories obtained for

Brody Dylan various spins, speeds, and release angles. Johnson

18 Introduction 16 Bowling Strikes 14 Underlying Physics 12

Mathematical 10 Model 8 Human Experiment 6

0 0 0.5 1 1.60 degrees

Brody Dylan Johnson Physics of Bowling 18

0 0 0.5 1 3.75 degrees

Cheating the Geometry

Physics of Bowling The model allows us to compare trajectories obtained for

Brody Dylan various spins, speeds, and release angles. Johnson

18 18 Introduction 16 16 Bowling Strikes 14 14 Underlying Physics 12 12

Mathematical 10 10 Model 8 8 Human Experiment 6 6

4 4

2 2

0 0 0 0.5 1 0 0.5 1 1.60 degrees 5.38 degrees

Brody Dylan Johnson Physics of Bowling Cheating the Geometry

Physics of Bowling The model allows us to compare trajectories obtained for

Brody Dylan various spins, speeds, and release angles. Johnson

18 18 18 Introduction 16 16 16 Bowling Strikes 14 14 14 Underlying Physics 12 12 12

Mathematical 10 10 10 Model 8 8 8 Human Experiment 6 6 6

4 4 4

2 2 2

0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 1.60 degrees 5.38 degrees 3.75 degrees

Brody Dylan Johnson Physics of Bowling Straight balls: Throwing Method Averge Strike % straight with a house Straight 120 17% ball. No-thumb 130 22% No-thumb spin: Creating Fingertip 160 27% spin by leaving the thumb out and torquing a house ball. Fingertip Grip: Creating spin with a personalized, ﬁngertip grip ball.

Good bowlers seem to get on the order of 50% strikes.

The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Method Averge Strike % Straight 120 17% No-thumb 130 22% No-thumb spin: Creating Fingertip 160 27% spin by leaving the thumb out and torquing a house ball. Fingertip Grip: Creating spin with a personalized, ﬁngertip grip ball.

Good bowlers seem to get on the order of 50% strikes.

The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction Straight balls: Throwing Bowling Strikes straight with a house

Underlying ball. Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Method Averge Strike % Straight 120 17% No-thumb 130 22% Fingertip 160 27%

Fingertip Grip: Creating spin with a personalized, ﬁngertip grip ball.

Good bowlers seem to get on the order of 50% strikes.

The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction Straight balls: Throwing Bowling Strikes straight with a house

Underlying ball. Physics No-thumb spin: Creating Mathematical Model spin by leaving the thumb out and torquing Human Experiment a house ball.

Brody Dylan Johnson Physics of Bowling Method Averge Strike % Straight 120 17% No-thumb 130 22% Fingertip 160 27%

Good bowlers seem to get on the order of 50% strikes.

The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction Straight balls: Throwing Bowling Strikes straight with a house

Underlying ball. Physics No-thumb spin: Creating Mathematical Model spin by leaving the thumb out and torquing Human Experiment a house ball. Fingertip Grip: Creating spin with a personalized, ﬁngertip grip ball.

Brody Dylan Johnson Physics of Bowling Good bowlers seem to get on the order of 50% strikes.

The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction Straight balls: Throwing Method Averge Strike % Bowling Strikes straight with a house Straight 120 17%

Underlying ball. No-thumb 130 22% Physics No-thumb spin: Creating Fingertip 160 27% Mathematical Model spin by leaving the thumb out and torquing Human Experiment a house ball. Fingertip Grip: Creating spin with a personalized, ﬁngertip grip ball.

Brody Dylan Johnson Physics of Bowling The Three Stages

Physics of Bowling The three trajectories on the previous slide illustrate three Brody Dylan Johnson stages of my bowling game:

Introduction Straight balls: Throwing Method Averge Strike % Bowling Strikes straight with a house Straight 120 17%

Good bowlers seem to get on the order of 50% strikes.

Brody Dylan Johnson Physics of Bowling The Three Stages

Physics of Bowling

Brody Dylan Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Straight Bowling

Physics of Average First-Ball Bowling 40 games: Brody Dylan 120 7.1 Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling No-Thumb Spin Bowling

Physics of Average First-Ball Bowling 100 games: Brody Dylan 130 7.6 Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling Fingertip Grip

Physics of Average First-Ball Bowling 60 games: Brody Dylan 160 8.4 Johnson

Introduction

Bowling Strikes

Underlying Physics

Mathematical Model

Human Experiment

Brody Dylan Johnson Physics of Bowling The dynamics of the bowling ball, the oil, the gutters, etc. exhibit interesting concepts from math and physics that are accessible to anyone with a background in diﬀerential equations.

Conclusion

Physics of Bowling

Human Experiment

Brody Dylan Johnson Physics of Bowling Conclusion

Physics of Bowling

Brody Dylan Johnson Bowling, like many sports, is enjoyable for players of all Introduction skill levels because it is easy to get started and see real Bowling Strikes results (like bowling a strike), yet most people can play for Underlying years without ever coming close to the elusive perfect Physics game. Mathematical Model The dynamics of the bowling ball, the oil, the gutters, etc. Human Experiment exhibit interesting concepts from math and physics that are accessible to anyone with a background in diﬀerential equations.

Brody Dylan Johnson Physics of Bowling