Speed Calculations Science Behind the Sport West Virginia University

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SCIENCE BEHIND THE SPORT
Speed Calculations | Science Behind the Sport | West Virginia University
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Speed Calculations
How to Calculate your Maximum Velocity on a Zip Line
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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.
Figure 2: Different geometries formed by the zip line cable hanging between two endpoints
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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
2
a
2
+ b
2
= c
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.
Figure 3: To find a and b respectively, find the two shortest sides of the triangle. Generally
c, the hypotenuse, will be the longest side of the triangle.
a
2
+ b
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2
= c
2
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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
2
f
= V
2
i
+ 2 × a × d
Where:
Vf
Vi
is final velocity
is initial velocity
a is acceleration
d is displacemen
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
Figure 4. 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.
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The distance traveled (the hypotenuse of the right triangle) is approximated using the
Pythagorean Theorem.
−−−−−−−
−−−−−−−−−−−−
−−−−−−−−−−
−
Distance(d) = √horizontal
2
distance
+ vertical
2
drop
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) =
Σ 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.
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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.
Figure 5: 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) =
Σ Forces
mass
m×g×sin(θ)−loss
=
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:
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Acceleration(a) = g × sin (θ) − loss
Step 3: Calculate Maximum Velocity
The maximum velocity can be determined using a kinematic equation:
Maximum
2
2
velocity (V max ) = initial
2
velocity
+ a × d
Where:
a = acceleration
d = distance traveled
Since the initial velocity is assumed to be zero, this equation simplifies to:
Maximum
−
−
−
−
−
−
−
−
Velocity(V max) = √2 × a × d
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