> Collisions

# Collisions

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### Introduction

The USBC specifies the geometry of a bowling pin by giving 15 diameters at 15 corresponding heights, as well as a radius for the hemispherical top of the pin.  It is interesting to note that the mass center of the pin is higher than the widest part of the pin, and the center of the ball is very near the widest part of the pin.  The 10 pins are arranged in equilateral triangles at the end of the lane, with the center of the nearest pin being 60 ft from the foul line.  It is universally accepted in the bowling community that an ideal right-handed throw brings the ball to the 1-3 pocket such that the 1 pin travels in a straight line toward the 7 pin.  In turn, the 3 pin should go toward the 10 pin, as shown below.

### 2D discs with friction

The following video shows a simulation of 2 dimensional collisions between a ball and 10 pins, all modeled as discs of unique radii, mass, and moment of inertia.  Contact is detected by comparing the distance between the centers of 2 objects to the sum of their radii:

$$\sqrt{(x_2 - x_1) ^2 + (y_2 - y_1) ^2} < r_1 + r_2$$

Post impact velocities are computed using an impulse-based model (see Stronge, 2000).  In this model, each pair of objects has the same coefficient of restitution and the same coefficient of friction.  There is no friction between the bodies and lane.

### 3D ball and pins without friction

The following video shows a simulation using pseudo-bounding spheres and sphere-swept surfaces to detect collisions.  See the presentation and videos in Collision detection for more on collision detection.  The post-impact velocity is 2D and frictionless.  The bodies are all given initial angular velocities and slight vertical offsets to make this a more challenging test of the collision detection algorithm used.  Only the headpin (red) is given an initial linear velocity.

### Collision dynamics results

From a bowler's perspective, ball position and entry angle at first impact with the pins are the 2 most important factors affecting pin "action" or pinfall. Ball mass should also be important, and with infinite ball speed the pins would probably bounce back and forth between the walls so many times they would all have to fall. First, we simultaneously examine the effects of position and entry angle for a baseline shot.  Using the 2D model, pinfall and which pins fell were recorded for 1600 simulations - using 40 positions from just grazing the right side of the headpin to hitting the headpin head-on, and 40 entry angles ranging from 0 to 7 degrees.  5 or 6 degrees is about as much entry angle as is attainable in practice.  In the figure below, white corresponds to 10 pins felled, the one black square in the upper left is 5, and shades of gray are 6-9.

The conclusions are summarized by addressing bowling myths:

• More entry angle yields greater pinfall.  Mostly true.
• More entry angle yields more splits.  True.
• More entry angle is favorable for light hits, and less entry angle is better for high hits.  Mostly true.
• More mass yields greater pinfall.  Mostly true.
• More massive balls deflect less and leave more 9-pins.  Mostly false.