Calculations: Procedure

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,

$$Acceleration (a) = \frac{\sum{Forces}}{mass}$$

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.

Graphic explaining force due to gravity

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:

$$Acceleration~(a) = \frac{\sum{Forces}}{mass} = \frac{m \times g \times sin(\theta) - loss}{m}$$ where: m = mass, g = gravitational acceleration, and θ = cable slope angle

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

$$Acceleration~(a)= g × sin(\theta) - loss$$


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