One aspect of billiards that we researched was the 90 degree rule. This rule states that when the cue ball strikes an object ball with no topspin or bottom spin, the two balls seperate at a 90 degree angle. Tangent line perpendicular to the impact line. (write all the specifics from the book)

To test this rule, we aimed the cue ball from a fixed spot 50 cm from the object ball at different angles. We hit the cue ball with no spin and tested hitting the ball soft or hard. We printed a protractor out to put on the table under the object ball, so we could determine the angle at which the balls seperated. From these results, we could determine how exact the 90 degree rule is under normal conditions.

First we aimed from an angle of 15 degrees. We hit the cue ball with a soft stroke. We tested this three times, as we did with all the experiments, and the separation between the balls was 68, 79, and 90 degrees. The results were all about less than the 90 degrees which we expected. Then we hit the cue ball with a hard stroke. The separation between the balls was 92, 90, and 92 degrees. This time our results matched up a lot better with what we excepted. I think this was because when we hit the ball softer, it was able to develop spin more easily, but when we hit it harder, the cue ball wasn’t able to develop as much spin.

Next we aimed at an angle of 45 degrees. When we hit the cue ball with a soft stroke, the differences were 72, 75, and 79 degrees. Again, our results for slow shots were less than what we expected. For a harder stroke, our results were differences of 82, 84, and 77. These results seemed to vary slightly, but they were again closer to 90 degrees than the slow shots.

Last, we aimed at the object ball at an angle of 75 degrees. The differences in angles for the soft shots were 56, 70, and 78 degrees. From this data, we noticed a trend that as the angle we aimed at the cue ball great larger, the difference in angle of separation seemed to be getting smaller for soft shots. For the fast shots, our results were 81, 80, and 77. Just like the soft shots, the angle of separation seemed to be getting lower as we aimed at the object ball at a greater angle.

The 90 degree rule states that the cue ball is not supposed to have any spin for the rule to be appliable, but we wanted to test the effects of topspin and bottom spin on the paths of the balls. So we did all the preceding tests again with topspin and bottomspin added.

Again, we aimed at the object ball at a 15 degree angle. We hit the cue ball softly, but this time with topspin. The differences in angles were 78, 82, and 79 degrees. These results were slightly less than hitting the cue ball with no spin, which is what we expected. When we hit the cue ball harder, the results were 99, 87, and 87 degrees. Except for our first piece of data in this set, the results were slightly lower than before.

Next we aimed at the object ball at a 45 degree angle. Again, we hit the cue ball softly with topspin. The resulting angles were 71, 76, and 80 degrees. These results were very comparable to when we hit the cue ball with no spin at this angle. Then we hit the ball with topspin again, only with a harder stroke. The angles resulting were 91, 85, and 84 degrees. These results were greater than when we hit the ball with no spin, but as in the other set of data, the resulting angles were getting lower as the angle we aimed at the object ball increased.

Lastly, we aimed at the object ball with a 75 degree angle with topspin. For the slow stroke, our results were 43, 54, and 45 degrees. This was a very big jump from before. The resulting angles were about half of what they should have been without topspin. Then we hit the cue ball with a harder stroke. The resulting angles were 60, 58, and 60 degrees. These results followed the trends from before; the harder shot created a bigger angle than the softer shot, and the resulting angle became less as the angle we aimed grew.

Finally to conclude this experiment, we hit the cue ball with backspin. Again, we started out aiming at the object ball with a 15 degree angle. For the soft shot with bottomspin, the resulting angles were 96, 96, and 87 degrees. We expected the bottom spin to make the angle bigger, and this data followed that theory. Then we hit a harder shot and the resulting angles were 96, 90, and 86 degrees. These results did not really prove what we thought would happen. They were very comparable to the results of hitting the ball with no spin and topspin.

Next we hit the object ball at a 45 degree angle. For the soft shot with backspin, the resulting angles were 75, 81, and 78 degrees. These results were slightly larger than from the other sets of data. Then we hit the cue ball harder, and the results were 92, 83, and 85 degrees. These results were almost identical to the results from hitting the back with topspin, which shouldn’t have happened. They were however greater than the angles from hitting the cue ball with no spin, which is what we were looking for.

Last, we hit the object ball at a 75 degree angle. We hit a soft shot with backspin again, and the differences in angles were 50, 52, and 68 degrees. These angles were less than from when we hit the cue ball with no spin, which is not what we were expecting. These angles we larger than from hitting the cueball with topspin, however. Finally, we hit the cue ball with a hard stroke and backspin, and the results were 95, 86, and 85 degrees. These angles were greater than when we hit the cue ball with no spin, which is what we were hoping for.

From these experiments, we noticed a few trends. We noticed that as the angle we aimed at the object ball grew, the resulting angles became smaller. We also noticed that as we hit the cue ball faster, the resulting angles were bigger. Lastly, from the information we noticed that when the ball was hit with topspin, the resulting angles were smaller than when we hit the cue ball with no spin, and that when we hit the cue ball with bottomspin, the resulting angles were greater than when we hit the cue ball with no spin. There were some exceptions to these trend, but for the most part that data followed these observations.

There is also a proof to the 90 degree rule which we can explain. The proof takes into consideration that there is no spin on the cue ball and that the collision is perfectly elastic Due to our conditions not being able to be perfect, we couldn’t get the results to be exactly 90 degrees every time, but with this proof, we can show why the results should be 90 degrees every time.

Another aspect of billiards that we tested was the 30 degree rule. The 30 degree rule states that when the cue ball hits an object ball with normal roll close to a half-ball hit, the cue ball deflects 30 degrees from the aiming line. (put more detail from the book)

To test this rule, we hit the object ball with a half hit, one quarter hit, and three quarter hit, then determined the angle at which the cue ball deflected from its aiming line. We aimed the cue ball 50 cm away from the object ball and hit it to produce normal roll. To figure out the angle at which the cue ball deflected, we placed the object ball on top of a printed protractor and measured the angle after the balls hit.

First we did a half ball hit with a slow stroke. The resulting angles were 27, 33, and 28 degrees. These results were right on with what we hoped for. Then we hit the cue ball with a harder stroke, and our results were not nearly as good. The resulting angles for the harder stroke were 62, 62, and 53 degree. These angles were a lot bigger than what they should have been. Looking back, I think that may have been because we didn’t hit the cue ball with the right roll.

Next we did a three quarter ball hit with a slow stroke. The resulting angles were 49, 52, and 50 degrees. Again, our results were much bigger than what the should have been. Then we did a faster stroke and the resulting angles were 65, 70, and 64 degrees. We had a lot of error again.

Lastly, we did a quarter ball hits. For the soft stroke, the resulting angles were 37, 36, and 37 degrees. These results were better and closer matched what we were looking for. For the fast stroke, the results were 46, 40, and 48 degrees. Again the results were big, but these weren’t as big as before.

From these results, I think its safe to say the 30 degree rule is a lot more prone to error than the 90 degree rule. It needs a lot more perfect conditions in order to be relevant to playing a normal game of billiards. There is also a proof for this rule which explains why this rule works under favorable conditions. It shows what the maximum and minumum cut angles should be for the half, quarter, and three quarter hits.

### Data

Distance from cue ball to object ball: 50 cm

No Spin:

Aiming Angles | 15 Degrees | 45 Degrees | 75 Degrees |
---|---|---|---|

Resulting Angles (Soft hit) | 68, 79, 90 | 72, 75, 79 | 56, 70, 78 |

Resulting Angles (Hard hit) | 92, 90, 92 | 82, 84, 77 | 81, 80, 77 |

Top Spin:

Aiming Angles | 15 Degrees | 45 Degrees | 75 Degrees |
---|---|---|---|

Resulting Angles (Soft hit) | 78, 82, 79 | 71, 76, 80 | 43, 54, 45 |

Resulting Angles (Hard hit) | 99, 87, 87 | 91, 85, 84 | 60, 58, 60 |

Back Spin:

Aiming Angles | 15 Degrees | 45 Degrees | 75 Degrees |
---|---|---|---|

Resulting Angles (Soft hit) | 96, 96, 87 | 75, 81, 78 | 50, 52, 68 |

Resulting Angles (Hard hit) | 96, 90, 86 | 92, 83, 85 | 95, 86, 85 |

Distance from cue ball to object ball: 50 cm

(All shots hit with normal roll)

Type of Hit | Quarter | Half | Three Quarter |
---|---|---|---|

Resulting Angles (Soft hit) | 37, 36, 37 | 27, 33, 28 | 49, 52, 50 |

Resulting Angles (Hard hit) | 46, 40, 48 | 62, 62, 53 | 65, 70, 64 |