Calculations: Background

Think of the zip line as a catenary curve shown in Figure 2. The catenary curve is the natural shape that a cable assumes hanging under its own weight when supported at the two ends. The cable of the zip line approximates a skewed catenary curve because of the different heights of the end points. When the rider's weight is applied to the cable, the shape of the cable will change as the rider's weight pulls the cable into a "straight line." The shape of the catenary curve now resembles more closely the lines of a triangle where the angles and side lengths reflect cable tension and the change in the rider's position.

Chart showing an idealized catenary curve (uniform curve between two points of equal height), a skewed catenary curve (curve between 2 points of unequal heights and skews toward the lower point), and the transformation of the curve into two triangles when weight hangs off line.

Figure 2: Geometries formed by the zip line cable hanging between two endpoints

The equations needed to calculate a rider’s theoretical maximum speed are derived from the Pythagorean Theorem, Newton’s Second Law of Motion, and a Kinematic Equation that relates displacement, velocity, and acceleration.

The Pythagorean Theorem relates the three sides of any right triangle – a triangle in which one angle is 90 degrees. The theorem states that ; in words it states that the square of the hypotenuse (the side opposite the 90 degree angle) is equal to the sum of the squares of the other two sides.

Diagram of Pythagorean Theorem

Newton’s Second Law of Motion states that the force of an object is equal to its mass multiplied by its acceleration. This verbal statement is often expressed in equation form as: F = m * a.

Kinematic Equations are a set of scientifically accepted equations for finding unknown information about an object’s motion when other information is known. The equations can be used for any motion that can be classified as either constant velocity or constant acceleration. The equation needed to find maximum velocity is as follows:

Where:
vf is final velocity
vi is initial velocity
a is acceleration
d is displacement

 

Upon stepping off the platform Newton's Laws of Motion become evident. The absence of the platform provides the unbalanced force that sets the rider in motion. The rider drops, gravity takes over, and you begin to accelerate. This illustrates the 2nd Law. The rider is no longer at rest and continues along the zip line until a braking force is applied.

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