In this section, we explore the path of a billiard ball as it travels around a rectangular pool table and determine which pocket the ball will land in. Throughout this section, we assume that there is no spin on the ball so that it travels in a straight line; it encounters no bumps, fuzz, or other impediments as it travels. We also assume that the ball has no resistance and will continue to travel until in lands in one of the corner pockets. Once again, we will see a connection between geometry and properties of the set of counting numbers.
Check the ‘wall or mirror’ box to show the wall that will deflect the ball and the angle this wall makes with the path of the ball. What is the measure of angle \(\angle{DBE}\text{?}\) Use geometric principles to explain how you know this.
Then check the ‘reflection’ box to show the path the ball will take after it leaves the wall. Point \(D'\) is the reflection of point \(D\) across line \(\overleftrightarrow{CE}\text{.}\) Explain why \(\angle{D'BE}\cong\angle{DBE}\text{.}\)
In the field of Optics, the angle of incidence is defined as the angle that the directional ray makes with the ray perpendicular to the mirror. Check the ‘perpendicular’ box to see the perpendicular \(\overline{BF}\) to line \(\overleftrightarrow{CE}\text{.}\) What is the measure of \(\angle{ABF}\text{,}\) the angle of incidence?
In Optics, angle \(\angle{D'BF}\) is called the angle of reflection. How is \(\angle{D'BF}\) related to \(\angle{ABF}\text{?}\) Give a thorough argument of why this must be the case for any acute angle of incidence.
Whether we focus on the angles that the incoming and departing rays make with the mirror or if we, like the physicists, compare the angles those rays make with the perpendicular to the mirror, the pair of angles are congruent. Keep this in mind as you construct your billiard ball paths.
Figure 6.2.5 shows the path of a billiard ball hit at a \(20^{\circ}\) angle from the lower left-hand corner and landing in the lower right corner pocket.
Suppose a ball hits each side at a \(45^{\circ}\) angle and continues to roll until it reaches a corner of the table. In the next exploration, we discover that the final corner is determined by the side lengths.
In our simplified game, we assume that the table has only four pockets, one in each corner and there is no spin on the ball. Whenever the ball meets a side, its path makes a \(45^{\circ}\) degree angle with that side. Our ball encounters no friction or interference, continuing along this path until it lands in one of the table’s corners.
For each of the pool tables that follow, sketch the path of a billiard ball shot from the lower left corner at an angle of \(45^{\circ}\) until it reaches a corner pocket. Record the pocket that the ball eventually reaches. Also record the number of hits the ball makes against the sides of the table. Include the starting corner and the ending corner in this total of hits.
