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I'm currently brainstorming for my Math AA HL Internal Assessment (a kind of math investigation paper you must elaborate for the IB Diploma) and I've hit a bit of a roadblock. Snowboarding is a hobby I'm incredibly passionate about, and my teacher suggested leveraging this interest for my IA to enhance personal engagement. The idea I've settled on is to calculate the feasibility of a quintuple cork in snowboarding.

This involves considering constant initial conditions such as the snowboarder's mass, the ramp and landing angles, etc. My initial thought was to apply basic kinematics of projectile motion, but I wanted to dive deeper into more complex mathematics. That's where Euler's equations for rigid body dynamics come in, which involve matrices and differential equations of rotation. These equations seem perfect for adding a sophisticated and challenging element to my project.

However, I'm facing a significant hurdle. The research papers and resources I've found on Euler's equations are predominantly collegiate-level. With no prior experience in these types of equations, I'm struggling to grasp the concepts and apply them to my project.

Has anyone here tackled something similar or have experience with Euler's equations in the context of a high school level project? I'm looking for advice or resources that could make these equations more approachable for someone at my level. Any help or guidance would be immensely appreciated.

Link to my original question (I decided to make another post for more visibility, since my other question is already old and doesn't obtain any further answers): Equations modelling a snowboarder/skier on a jump

Miles Jarra Gloekler
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One problem you might have is that Euler's equations relate external torque to angular acceleration and apply to a rigid body. Some of the torque might come from the curved shape of the ramp. But some of it comes from twisting your shoulders and leaning back as you leave. For a quintuple twist, you will need a vigorous twist. Also you control your rate of spin by tucking or moving your arms.

This means you aren't rigid. That isn't fatal. But it means your moment of inertial changes from moment to moment.

All the external torque come from the force of the snow on the board. Once you are airborne, the torques are $0$. This doesn't mean you can't change angular velocity. It means you do so by changing your moment of inertia.

mmesser314
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  • The model isn't meant to be perfect, as I'm assuming that would be way harder to replicate, rather I would assume the snowboarder to be a cylinder that rotates at different rates, hence the differential equation (since the rotation when in the air isn't constant). Nevertheless, are you meaning to say I should model the motion with another set of equations? If so, which would you reccomend? – Miles Jarra Gloekler Nov 24 '23 at 20:28
  • A rigid cylinder in air will rotate at a constant rate. The Euler equations will tell you about how the cylinder starts rotating in response to forces the snow exerts on the board. They also can tell you how the rotation rate will change when you change your shape, like an ice skater spinning faster when she pulls in her arms. – mmesser314 Nov 25 '23 at 00:59
  • So, considering I am trying to model on a jump, what would you suggest I do? Should I try to find another set of equations? And if I am able to stick to Euler's, how should I go about changing the shape of the cylinder? Making the radius wider? – Miles Jarra Gloekler Nov 25 '23 at 19:19
  • If you were doing a high jump project, you would use $F = ma$ and $d= d_0 + v_0t + 1/2at^2$. You would find out how much upward force the athlete exerted to find $a$, You would use that to figure out the height. – mmesser314 Nov 25 '23 at 21:33
  • Your project will be like the high jump, but for rotation. Euler's equations are the rotational equivalent of $F=ma$. You need to find the rotational equivalent of $d$. So think about how you would model a high jump. The high jump is easier because it is in $1$ dimension. Once you understand it, move on to the more complex rotational problem. – mmesser314 Nov 25 '23 at 21:37
  • I have tried solving the high jump example with variables (https://docdro.id/3Pmm2Ic), which can be found in the following PDF, I am not looking for a correction as if you were my teacher, since you are being of great help and I don't want to overstep anything. Yet, I still do not understand how to apply Euler's equations to the problem at hand. Could you provide some insightful resources if possible? – Miles Jarra Gloekler Dec 05 '23 at 20:31
  • I think the problem is that you are thinking about equations and how to manipulate them. You should be thinking about physics. What are the forces the ramp exerts on the snowboarder and how do they lead to rotations? How does the motion of the snowboarder as he launches generate those forces? While he is in the air, his shape determines his moment of inertia, and that determines his angular velocity. What is the shape of the snowboarder, and how does it change? What is his moment of inertia? Once you know these things, you can use them in the Euler equations. – mmesser314 Dec 06 '23 at 00:24
  • It is a pretty complex problem. In part because you need to learn how to think about all this physics. That alone is a big task. I might suggest trying it again after you finish the first year of college physics classes. But here is a post that will get you started thinking about how forces lead to torques, which lead to rotations. Toppling of a cylinder on a block – mmesser314 Dec 06 '23 at 00:28