View book Table of Contents

Order book

Links

Contact the author.

Algebra for Athletes.com             Excerpts from the book Algebra for Athletes,

Cameron Bauer, 2008, Nova Science Publishers, Inc.

Algebra for Athletes is the #1 bestseller at Nova Science Publishers

12. Cyclical Motion

In the section on Speed we learned to understand linear motion by breaking it down into components of distance, speed, and time.  In the section on parabolas, we learned how to understand parabolic motion by breaking it down into horizontal velocity, vertical velocity, and vertical acceleration.  Another type of motion thatís often found in sports is cyclical motion. 

Running and swimming involve cyclical motions.  When dribbling down the court, a basketball player moves her hand in sync with the motion of the ball.  To develop quickness, a boxer punches a hanging bag in sync with the motion of the bag. An athlete jumping rope moves the rope with his hands and jumps in a particular rhythm to miss the rope.  School children who play double dutch jump to an even more complex rhythm. 

Cyclical motion occurs in countless other areas in the physical world.  The pistons in an engine will fire in a precise sequence thousands of times in a minute.  A mechanic will check the engine using a flashing timing light to make sure it's running properly.  Electrical motors spin running everything from pumps to power tools.  A radio station will broadcast radio waves which, when hitting an antenna attached to a tuned-in stereo, will vibrate a speaker creating music.   Similarly, a television receives electromagnetic waves that cause a stream of light to hit thousands of dots on a screen millions of times per second to create a picture.

Like many of the examples in sports, the operation of machinery involves the motion of two or more objects which must move perfectly together in order to work. The basketball player's hand must move perfectly with the ball and the pistons in an engine must fire perfectly in order to turn an axle.  Making things move together properly requires a thorough understanding of cyclical motion.  The same math tools are used for all types of cyclical motion from that of a motor to the cycles of a TV or computer screen. 

As one might expect, the math involved in cyclical motion is based on circular motion. Once the mathematics of circular motion is understood, it can be expanded to understand virtually any other type of cyclical motion.

To examine circular motion, let's take a closer look at an athlete skipping rope. To monitor the motion of the rope, we'll place an xy plane on the athlete and put his hands at the origin.  The ends of the rope (his hands) remain in a relatively steady position at the origin.  The tip of the rope extends 1 meter in the plane of the graph.  The rope remains relatively straight as it rotates.  The rope rotates counterclockwise and just barely touches the ground.

Skipping rope requires keeping track of the location of the tip of the rope.  The athlete must jump when the tip of the rope is near his feet.  The position of the tip can be described by either: 

            1. The angle of the rope, , or

            2. The height (y-value) of the tip above or below the x-axis.

In the figure above, the rope has rotated to an angle .  A triangle is created which helps keep track of the position of the tip of the rope.  The leg of the triangle which represents the height of the tip above the x-axis is marked with an o, for the length of the leg opposite the angle.  The other leg is marked with an a for the length of the leg adjacent to the angle.  The hypotenuse of the triangle is marked with an h.  Analyzing cyclical motion involves understanding the relationships that exist between  and the lengths of the sides of the triangle.  For a right triangle of a certain shape, the relationships between  and the lengths of the sides remain constant regardless of its size.  Because of the triangle that is formed in the figure, the branch of mathematics which studies the relationship between the angles and sides is known as trigonometry.

12.1.1 The Sine Function 

A very important relationship in math is found by relating  and the ratio of o and h. This relationship is known as the sine of .  This relationship is expressed sin .  The sine function is given by the following equation:

                                 (12-1) 

In the jump rope example, the length of 1 meter to the tip of the rope was chosen for a particular reason.  Since the discussion states that the tip of the rope remains 1 meter from the athlete's hands, the length of the hypotenuse is always 1.  Recall that any real number divided by 1 equals the number.  If h is always equal to 1, the sine of the angle will always be equal to the length of the opposite leg.  In mathematical terms:

In trigonometry, the circle with a radius of 1 is known as the unit circle. Click the equation above to see how the unit circle is used to show the graph of the sine function. The figure shows how the height of the opposite leg changes with .  The curve that is created is the graph of y = sin .  This kind of curve is known as a sine wave.

Another important trigonometric relationship compares  with the adjacent leg of the triangle.  This relationship is known as the cosine. It also has a graph like a sine wave but since the adjacent leg starts out with a length of 1 when  = 0, the plot of the cosine has its crest at  = 0.

Another important function is the tangent function.  The tangent function is the ratio of the opposite leg over the adjacent leg. 

Using Trigonometry to Find Distances

The Pythagorean Theorem provided one method of finding the unknown length of a side of a right triangle when the lengths of the other two sides were known.  By using the trigonometric functions, we can determine the lengths of the other two sides by knowing the length of one of the sides and the size of one of the angles.  The following problem shows how to use the sine function to calculate distances. 

Example 12.1:  A quarterback throws a pass that travels 20 yards  to his wide receiver.  The pass is thrown at a  angle with the yardage markers.  How far downfield from the quarterback is the receiver?

 Solution:  We notice that the distance of the pass represents the hypotenuse of a triangle.  The distance downfield, d, represents the length of the opposite leg of this triangle.  From Eq. 12-1: 

                                    Therefore            whered = the distance downfield from

                                                                                                        the quarterback

            Solving the equation for d,             d = 20sin

By using a calculator we can find that sin = 0.5

                        So:                                  d = 20(0.5) 

                                                                = 10 yards 

The pass travels 10 yards downfield.