
Danse Macabre. Credit to Wikipedia.The Euthanasia CoasterWith Halloween having just passed yesterday, I thought I would make a festive post. Before the candy and costumes Halloween was a time recognized for when the world of the dead and the world of the living intermixed and the dead would walk among us. Such morbid thoughts have inspired my blog about the roller coaster guaranteed to thrill you to death.The roller coaster was designed by Julijonas Urbonas (http://www.julijonasurbonas.lt/p/euthanasiacoaster/) with the idea of creating an alternative future form of euthanasia to showcase how death can be taken out of "churches and shrines" and placed in a more contemporary setting like an amusement park. However interesting it is, the purpose of this post is to explore the physics of such a design and not the ethical and moral implications of adopting this as a form of euthanasia. 
1:1000 scale drawing of Euthanasia coaster. Credit Urbonas. 
So, how does the euthanasia coaster kill you? Mainly, the death shall be caused through cerebral hypoxia or having the brain's oxygen cut off. There are two elements as Urbonas explains.
500 meter drop. Credit Urbonas. 
First, there is the 500 meter drop height that brings you close to terminal velocity by the time you reach the bottom. Some people who have viewed the exhibit say that the two minute ascension to the top might just cause death by fright. To put it in perspective, you'll be nearly onethird of a mile up. This is about the height of the Sears "Willis" Tower and over 50 meters higher than the Empire State Building measured from their very tips.
We can calculate the velocity just before train reaches the bottom through the conservation of total energy equation.
You're much higher than this. (Skydeck at Willis Tower) 
We can calculate the velocity just before train reaches the bottom through the conservation of total energy equation.
∆ KE = ∆PE + Wnc
I assume that the frictional force is relatively very low. This assumption is justified because of the steepness of the drop reducing the normal force between track and coaster wheels and the low coefficient of friction possible on coasters today (0.0090.018 according to coaster101.com). An aerodynamic design and the relative mass of the roller coaster compared to the front of it can justify our ignoring of drag force. We can expect advanced designs and materials to go into the coaster as it is the last ride of someone's life.
As such, I can solve for the final velocity at the bottom.
∆ KE = ∆PE
0.5*velocity^2 = g*drop height
velocity = 98.9 m/s
= 100 m/s after SF
Using kinematics, we can see how long it takes use to drop 500 meters.
vf = vi + a*t
98.9 m/s = 0 + 9.8 m/s^2 * t
t = 10.1 sec
= 10 sec after SF
Now this is rather thrilling, isn't it? The world's current tallest roller coaster is the Kingda Ka with a drop of 127 meters. On the Euthanasia Coaster, we are dropping nearly four times that height for 10 seconds.
Now, the 500m drop is not life threatening unless you're overly excitable. The part that will end your life are the seven loops after the drop.
At the bottom the rider is going at a speed about 100 m/s and they immediately begin to enter a series of inverted teardrop shaped loops. These are special loops called clothoid loops that lower the entry speeds necessary to complete a loop. These loops as you will notice in the picture spiral down in size. This is done on purpose to compensate for the now signification friction slowing down the coaster.
As you will remember, going around a loop gives rise to a radial acceleration directed towards the center of the loop. The radial acceleration you experience would always be pointed towards the center of the loop. To look at it from your perspective, the acceleration always points from your feet to your head. However, your body has inertia and much like suddenly accelerating forward in a the car makes you feel like you're being pushed backwards into the seat, the radial acceleration makes you feel like you're being pushed by a force in the opposite direction of the radial acceleration. That is, you'd feel it from your head pushing to your feet. This causes blood to be pushed away from your head and into your lower body.
The loops are designed so that the radial acceleration is a constant 10g. This means that the force pushing the blood away from your head will be 10 times your weight on earth. Each successive loop decreases in radius to compensate for the decreasing velocity.
Human beings can handle extreme acceleration for short instances of time but the way that the Euthanasia Coaster is designed is to have you experience a high, but not painfully so, acceleration for a significant amount of time. The designer states that the loops are designed to take 60 seconds to complete.
So, for a full minute you experience 10g's. Your oxygen deprived brain experiences gLOC or gforce induced loss of consciousness from all the oxygen being pushed down to your legs. The designer says most people will be unconscious by the time they complete the first loop but there will remain some who may be resistant to g forces. However, no one is predicted to be able to finish the second loop awake. Death by oxygen deprivation is described as euphoric and gLOC can often cause strange and fanciful dreams.
We could easily approximate the loops by measuring the diagram on ... website but suppose we were tasked to create a series of 7 loops that would create 10g's of radial acceleration but whose radii would have to account for energy lost to friction. We are given the initial velocity, 98.9 m/s which was calculated earlier. We shall assume the mass of the coaster and riders is 500 kg and that the coefficient of friction is 0.018.
To find the rest of the loops something like the following must be done.
∆ KE = ∆PE + Wnc
∆ KE = Wnc => ∆ KE = Ffr*d
0.5*m*(vf^2  vo^2) = u*m*g*d => 0.5*(vf^2  vo^2) = u*g*d
0.5*vo^2 = u*g*d + 0.5vf^2 Equation 1
We have two unknowns d (circumference) and vf. However, we can create another equation between d and vf.
a = (vf^2) / r d = 2*pi*r
a*r = vf^2 r = d / (2*pi)
vf^2 = (a*d) / (2*pi)
vf = Sqrt [(10*g*d) / (2*pi)] Equation 2
Using both equations, we can set up a system of equations and solve for d.
d = (.5*vo^2) / [( (10 / (4*pi) ) + u)*g]
Notice that we do not require mass to calculate.
Taking that we can solve for vf with equation 2. We can also solve for radius by dividing d by 2*pi
The vf should then become the vo for the next loop and we shall have to find the succeeding vf and radius.
Because the calculations for all seven loops shall be tiresome, I have entered the formulas in Excel and allowed the program to calculate for me.
Now, the predicted time the coaster would spend on the loops is 60 seconds while according to this data it would be around 42 seconds. This discrepancy is due to ignoring of air resistance for if air resistance was factored in, loop design would have to change and so would velocities.
In the end, however shocking this Euthanasia Coaster may be, it can still be hacked with the help of a gsuit used in today's jets. I leave you with a simulation of the Euthanasia Coaster:
I assume that the frictional force is relatively very low. This assumption is justified because of the steepness of the drop reducing the normal force between track and coaster wheels and the low coefficient of friction possible on coasters today (0.0090.018 according to coaster101.com). An aerodynamic design and the relative mass of the roller coaster compared to the front of it can justify our ignoring of drag force. We can expect advanced designs and materials to go into the coaster as it is the last ride of someone's life.
As such, I can solve for the final velocity at the bottom.
∆ KE = ∆PE
0.5*velocity^2 = g*drop height
velocity = 98.9 m/s
= 100 m/s after SF
Using kinematics, we can see how long it takes use to drop 500 meters.
vf = vi + a*t
98.9 m/s = 0 + 9.8 m/s^2 * t
t = 10.1 sec
= 10 sec after SF
Now this is rather thrilling, isn't it? The world's current tallest roller coaster is the Kingda Ka with a drop of 127 meters. On the Euthanasia Coaster, we are dropping nearly four times that height for 10 seconds.
Series of seven clothoid loops. Credit Urbonas. 
Now, the 500m drop is not life threatening unless you're overly excitable. The part that will end your life are the seven loops after the drop.
At the bottom the rider is going at a speed about 100 m/s and they immediately begin to enter a series of inverted teardrop shaped loops. These are special loops called clothoid loops that lower the entry speeds necessary to complete a loop. These loops as you will notice in the picture spiral down in size. This is done on purpose to compensate for the now signification friction slowing down the coaster.
As you will remember, going around a loop gives rise to a radial acceleration directed towards the center of the loop. The radial acceleration you experience would always be pointed towards the center of the loop. To look at it from your perspective, the acceleration always points from your feet to your head. However, your body has inertia and much like suddenly accelerating forward in a the car makes you feel like you're being pushed backwards into the seat, the radial acceleration makes you feel like you're being pushed by a force in the opposite direction of the radial acceleration. That is, you'd feel it from your head pushing to your feet. This causes blood to be pushed away from your head and into your lower body.
The loops are designed so that the radial acceleration is a constant 10g. This means that the force pushing the blood away from your head will be 10 times your weight on earth. Each successive loop decreases in radius to compensate for the decreasing velocity.
Human beings can handle extreme acceleration for short instances of time but the way that the Euthanasia Coaster is designed is to have you experience a high, but not painfully so, acceleration for a significant amount of time. The designer states that the loops are designed to take 60 seconds to complete.
So, for a full minute you experience 10g's. Your oxygen deprived brain experiences gLOC or gforce induced loss of consciousness from all the oxygen being pushed down to your legs. The designer says most people will be unconscious by the time they complete the first loop but there will remain some who may be resistant to g forces. However, no one is predicted to be able to finish the second loop awake. Death by oxygen deprivation is described as euphoric and gLOC can often cause strange and fanciful dreams.
We could easily approximate the loops by measuring the diagram on ... website but suppose we were tasked to create a series of 7 loops that would create 10g's of radial acceleration but whose radii would have to account for energy lost to friction. We are given the initial velocity, 98.9 m/s which was calculated earlier. We shall assume the mass of the coaster and riders is 500 kg and that the coefficient of friction is 0.018.
To find the rest of the loops something like the following must be done.
∆ KE = ∆PE + Wnc
∆ KE = Wnc => ∆ KE = Ffr*d
0.5*m*(vf^2  vo^2) = u*m*g*d => 0.5*(vf^2  vo^2) = u*g*d
0.5*vo^2 = u*g*d + 0.5vf^2 Equation 1
We have two unknowns d (circumference) and vf. However, we can create another equation between d and vf.
a = (vf^2) / r d = 2*pi*r
a*r = vf^2 r = d / (2*pi)
vf^2 = (a*d) / (2*pi)
vf = Sqrt [(10*g*d) / (2*pi)] Equation 2
Using both equations, we can set up a system of equations and solve for d.
d = (.5*vo^2) / [( (10 / (4*pi) ) + u)*g]
Notice that we do not require mass to calculate.
Taking that we can solve for vf with equation 2. We can also solve for radius by dividing d by 2*pi
The vf should then become the vo for the next loop and we shall have to find the succeeding vf and radius.
Because the calculations for all seven loops shall be tiresome, I have entered the formulas in Excel and allowed the program to calculate for me.
Loop  Initial Velocity (m/s)  Final Velocity (m/s)  Circumference (m)  Radius (m) 
1  98.9  97.8  613  98 
2  97.8  96.7  600  95 
3  96.7  95.6  586  93 
4  95.6  94.6  573  91 
5  94.6  93.5  561  89 
6  93.5  92.5  548  87 
7  92.5  91.5  536  85 
Now, the predicted time the coaster would spend on the loops is 60 seconds while according to this data it would be around 42 seconds. This discrepancy is due to ignoring of air resistance for if air resistance was factored in, loop design would have to change and so would velocities.
In the end, however shocking this Euthanasia Coaster may be, it can still be hacked with the help of a gsuit used in today's jets. I leave you with a simulation of the Euthanasia Coaster:
I know this is a relatively old post, however, I would like to thank you for the thorough explanation of the physics behind such ride! What brings me here is that I actually read about this coaster somewhere I don't remember (maybe tumblr?) a while ago and now I'm taking physics 1 in college and we have to do a poster project and my brain just reminded me of this coaster as a really really interesting topic! We benefited a lot from the equations you provided as we couldn't easily figure out how to find the circumference of the loops before google brought us here :). We actually tried the "scaling method" manually and from the radii we got we wanted to calculate the circumference but then we were faced with the problem that the loops aren't really "circular". This leads me to the question, I saw in this link "http://www.physicsclassroom.com/class/circles/u6l2b.cfm" and now I'm wondering how does picture 3 (http://www.physicsclassroom.com/Class/circles/u6l2b3.gif) fit here? Meaning, are the radii and circumferences calculated for the actual loops or the three circles as outlined in the picture, three of which are hypothetical circles? At positions A, B and D the radii are larger than position C, therefore is the 10g applied within circle C only or is the centripetal acceleration really the same for the whole loop? If so then how? I realize that the velocity decreases at positions A and B which should further decrease the magnitude of centripetal acceleration, in addition to the direction of acceleration and centripetal acceleration being almost opposite at position A and almost perpendicular at position B, but the velocity increases again between positions C and D. In fact, you outlined this concept in explaining the way clothoid loops work. My questions simply put: Is the total centripetal acceleration during the whole loop 10g and that acceleration varies within the loop from position to position? If that's so, then what do the radii and circumferences calculated signify?
ReplyDeleteI apologize for the lengthy comment but truly this post deserves more talk than "No Comments"!