Tyler Gleckler
MAT 265 – Brewer
Honors Contract
M/W/F 12:00 – 12:50
An Ollie Down El Toro
Skateboarders are notorious for soaring through the air at remarkable speeds and down immense heights. Although it may appear that skateboarders are simply flinging themselves down stairs like ragdolls, their movements to achieve such feats are actually quite precise. Using Calculus, the secrets as to how these athletes do what they do will be revealed. A famous stair case located in California called “El Toro” that stands nearly 25 feet will be used to stress how impressive a skateboarders actions truly are, and more specifically, explain the motion of a skateboarder doing an Olly (jumping down a set of stairs) down El Toro.
Part I: Assumptions, Estimations, and Further Introduction.
When a skateboarder does an Olly he or she is technically a projectile, or will at least assumed to be for the purpose of this assignment. As a result, a series of equations derived by using Calculus can be used to explain the motion of the skateboarder. These equations are:
Equation 1: x(t) = vi*cos(θi)*t Displacement in x-direction
Equation 2:vix = vi*cos(θi) x-component of velocity
Equation 3: y(t) = vi*sin(θi)*t – (1/2)*g*t2 + hi Displacement in y-direction
Equation 4: viy = vi*sin(θi) – g*t y-component of velocity
Equation 5: x(t) = 8t x-position expression using assumptions
Equation 6: y(t) = 7*tan(45)*t – (1/2)*(9.81)*t2 + .915 y-position expression using assumptions
Equation 7: d = vi*t + (1/2)*a*t2 Distance as a function of time
Equation 8: vf = vi+ a*t Velocity as a function of time
Skateboarders of course range in heights, weights, and other factors that would affect the calculations done in this assignment, and for that reason some assumptions must be made in order to quantify the projectile (skateboarder). It will be assumed that:
- The skateboarder will be 1.83 meters tall (6 feet), with his/her center of mass being at .915 meters (3 feet).
- The skateboard will make an angle of 45 degrees with the horizon once the tail of the board makes contact with the ground.
- The initial speed of the skateboarder (vi) must be extremely fast, as El Toro is such a massive staircase, and thus the initial speed of the skateboarder will be assumed to be 8 meter/second (approximately 18 mph). It should be noted that before the skateboard begins to Ollie, his motion will be entirely in the x-direction.
- El Toro will be assumed to be exactly 7.62 meters.
Part II: Maximum Height, Final Velocity in the y-component and Slope of the “Steez” Line.
Maximum Height:
In order to determine the maximum height that the skateboarder is above the ground throughout his motion, the Calculus concept of a maximum will be used. To do this, first the derivative of Eq. 6 will be taken as the result is as follows:
7*tan(45) – 9.81*t = y’(t)
Next, the derivative will be set equal to 0 in order to find the critical numbers:
7*tan(45) – 9.81*t = 0
7*tan(45) = 9.81*t
7/9.81 = t
t = .714
Now, one number smaller than .714 and one number larger than .714 will be chosen and used as an input into derivative in order to determine intervals of increasing and decreasing to confirm whether or not this value is amaximum:
7*tan(45) – 9.81*(0) = 7
7*tan(45) – 9.81*(2) = -12.62
Being that numbers less than .714 are negative and values greater than .714 are positive, it can be said that at t = .714 there is a maximum. More specifically, from (-INF, .714) the function is increasing, and from (.714, INF) the function is decreasing.
With a maximum confirmed at t = .714, this critical number can be plugged into the original function in order to calculate the maximum height (not yet including the height of the stair) that the skateboarder is above the ground:
7*tan(45)*(.714) – (1/2)*(9.81)*(.714)2 + .915 =
7*.714 – 3.41 =
= 1.58 meters
This value is the height of the skateboarder above the ground relative to his starting height. To calculate the height the skateboarder is above the ground relative to the bottom of the staircase, the height of the staircase must be added to the calculated height:
1.58 + 7.62 = 9.20 meters
After .714 seconds from the Ollie, a skateboarder skating El Toro would be an immense 9.20 meters (approximately 30 feet) above the ground.
Final Velocity in the y-component:
Now that the maximum height of the skateboarder above the ground relative to the bottom of the staircase is known, that value can be used to calculate the final speed in the y-component. By deriving Eq. 7, Eq. 8 is determined. This is because the derivative of position is velocity, the derivative of velocity is acceleration, as so on:
d = vi*t + (1/2)*a*t2
vi*(1) + (1/2)*(2)*a*t
vf = vi +a*t
Alternatively: vf2 = vi2 + 2*a*d
With this equation, values must simply be plugged in, in order to obtain a final velocity in the y-direction. Final velocity is unknown, initial velocity (in the y-direction) is 0 as it is at its maximum, the acceleration (in only the y direction) is 9.81 m/s2, and the distance is equal to the height the skateboarder is above the ground which is 9.20 meters:
vf2 = 02 + 2*9.81*9.20
vf2 = 180.504 m/s
vf = 13.435 m/s
To put this speed into perspective, it would take less than 7 seconds to travel the length of a football field at this speed. Considering that this speed will abruptly turn to 0 m/s in the y-direction, consider the force applied to the skateboarders legs!
Part III: Area under the Curve of the Skateboarders Trajectory:
First using assumptions as well as values calculated in part II, the time required for the skateboarder to reach the bottom of the staircase is calculated as follows:
vf = vi + a*t
13.435 = 0 + 9.8t*t
13.435/9.81 = 1.370 = t
1.370 (time required to fall from the skateboarders maximum heights) + .714 (time required for the skateboarder to reach maximum height)= 2.084 seconds (Total time)
=
(7/2)*tan(45)*t2 – ((4.905)/(3))*t3 +.915*t =
(7/2)*tan(45)*2.0842 – ((4.905)/(3))*t3 + .915*2.084 =
2.309 m2was covered by the skateboarder during his trajectory.