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Calculations

 

Calculations: Overview

You can calculate maximum velocity using the following 3 steps:
  1. Determine the Distance Traveled
  2. Approximate Acceleration
  3. Calculate Maximum Velocity

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 a 2+b 2=c 2; 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. Pythagorean Theorem

a 2 + b 2 = c 2

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:

V f 2 = V i 2 + 2 × a × d

Where: 
v f is final velocity
v i 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 2 nd Law. The rider is no longer at rest and continues along the zip line until a braking force is applied.

Calculations: Procedure

Step 1: Determine the Distance Traveled

Zip line diagram with the distance, vertical drop, and horizontal distance to the lowest point labeled.  

Figure 3. Zip Line Diagram with the distance traveled, vertical drop, and horizontal distance to the lowest point of the line labeled. Assume the hypotenuse of the right triangle approximates the line of travel of the rider to the lowest point on the zip line. 

The distance traveled (the hypotenuse of the right triangle) is approximated using the Pythagorean Theorem.

D i s t a n c e ( d ) = h o r i z o n t a l   d i s t a n c e 2 + v e r t i c a l   d r o p 2

Step 2: Approximate Acceleration

We know that acceleration reflects Newton's Second Law of Motion (F = ma). Since there are multiple forces acting on the rider, the symbol epsilon 'Σ' must be used to represent the sum, or net, of these forces. Thus,

A c c e l e r a t i o n ( a ) = F o r c e s m a s s

In reality, there are two primary forces that affect acceleration; one is due to the component of gravity in the direction of motion and the other is a combination of friction and other resistive forces (collectively called 'loss').

On a zip line, friction and the other resistive forces are difficult to calculate. However, since these resistive forces have a significant impact on the calculated results (potentially causing up to a 1/3 reduction in maximum speed), they cannot be ignored. The best method for quantifying loss is through scientific experimentation.

Gravity acting on the rider's mass produces the dominant force. Consider the rider moving across the zip line as a variation of the classical block sliding down an inclined plane problem that is introduced in many entry level physics courses. We can apply the sine function of the cable slope angle (θ) to calculate the force due to gravity in the direction of motion. This can be better understood by observing Figure 4.

 
Free body diagram of a person riding a zip line. The zip line rider is represented by the gray box, the zip line cable is represented by the red diagonal line, and the force of gravity (weight) is shown in orange with its two vector components in blue. 
Figure 4: The zip line rider is represented by the gray box, the zip line cable is represented by the red diagonal line, and the force of gravity (weight) is shown in orange with its two vector components in blue.

Thus, the force due to gravity in the direction of motion is determined by multiplying the rider's mass (m) by gravitational acceleration (g) by the sin (θ).

The original equation now becomes:

A c c e l e r a t i o n   ( a ) = F o r c e s m a s s = m × g × s i n ( θ ) l o s s m

Where:
m = mass, 
g = gravitational acceleration
θ = cable slope angle


Since the mass term in each part of the equation cancels, the equation simplifies to:

A c c e l e r a t i o n   ( a ) = g × s i n ( θ ) l o s s


Step 3: Calculate Maximum Velocity

The maximum velocity can be determined using a kinematic equation:

M a x i m u m   V e l o c i t y 2   ( V m a x 2 ) = i n i t i a l   v e l o c i t y 2 + 2 × a × d

Where:
a = acceleration 
d = distance traveled


Since the initial velocity is assumed to be zero, this equation simplifies to:

M a x i m u m   V e l o c i t y ( V m a x ) = 2 × a × d