Problem 78 - Algebra and Trigonometry

Question

An elliptipool is an elliptical pool table with only one pocket. A pool shark places a ball on the table, hits it in what appears to be a random direction, and yet it bounces off the edge, falling directly into the pocket. Explain why this happens.

Step-by-Step Solution

Answer

The ball always falls into the pocket on an elliptical pool table due to the reflection law for ellipses and properties of scalar fields. Regardless of the direction the ball is hit, if a line is drawn from one focus (the initial position of the ball) and it reflects off the inside wall of the ellipse (the pool table), it will always pass through the other focus (the pocket).

Step 1: Understanding Ellipses

An ellipse is a type of conic section. It is defined as the set of all points such that the sum of the distances from two fixed points called foci (singular: focus) is constant. In our problem, consider the elliptipool as an ellipse and the solitary pocket and pool shark's original firing position as the two foci of ellipse.

Step 2: Reflection Law for Ellipses

In any ellipse, if a line is drawn from one focus and it reflects off the inside wall of the ellipse, it will always pass through the other focus no matter the angle it hits the wall. This property is due to the definition of an ellipse and properties of scalar fields, and this property is the basis for whispering galleries and elliptical rooms where a sound made at one focus can be heard clearly at the other focus.

Step 3: Application to the Elliptical Pool Table

The pool table is elliptical and if we consider the ball's trajectory from when it is hit, it bounces off the edges of the table and heads to the pocket. This is considered a reflection of the ball off the sides of the elliptical table. Due to the reflection law of ellipses, any line from one focus (in this case, the initial position of the ball when hit) will always pass through the other focus (the pocket) once it reflects off the wall; hence, the ball will always fall into the pocket, regardless of the initial direction the ball was hit.