Question

I am currently a highschool student elaborating an internal assessment for my mathematics AA HL course (a sort of research paper we must elaborate for the IB) and was looking into the relationship between mathematics and snowboarding. My current idea for research was investigating the feasibility of a quintuple cork given some initial conditions of the jump (angle, length, height, slope angle, etc.) and the rider (height, mass, etc.).

However, I've encountered some difficulties in finding appropriate differential equations to model a snowboarder's motion during a jump. Most existing models are either too complex, relying on matrix methods and other mathematics that are beyond my current level of math, or too simplistic, relying on basic kinematic equations that wouldn't allow for a meaningful investigation.

I would greatly appreciate your help in finding or developing differential equations that can accurately model a snowboarder's motion during a jump. I believe that with the right equations, I can utilize various numerical methods such as Euler and Runge-Kutta to analyze the feasibility of a quintuple cork on a specific jump. Whether you can suggest equations, recommend textbooks or research papers, or provide insights into the application of calculus and differential equations in this context, I would greatly appreciate your input.

Let me make it clear I do not want my work done for me, since it would be against guidelines, rather some pointers to help me get started

Answer 1

Modeling the motion of a snowboarder during a jump involves considering the forces and torques acting on the snowboarder, as well as the resulting motion.

Developing a realistic model for a snowboarder's jump, particularly a high-difficulty maneuver like a quintuple cork, requires a complex system of differential equations. While finding an exact solution can be challenging, various approaches can provide valuable insights and approximations.

1. Lagrangian Mechanics: This approach utilizes the principle of least action, where the snowboarder's motion minimizes the action integral. This leads to the Euler-Lagrange equations, providing a system of differential equations describing the motion under various forces and constraints.

2. Rigid Body Dynamics: Model the snowboarder as a system of interconnected rigid bodies (torso, legs, snowboard) with appropriate joints and constraints. Apply the laws of Newtonian mechanics (forces, torques) to derive differential equations governing the motion of each body.

Answer 2

To model a snowboarder's motion during a jump, you can use the equations of motion for projectile motion. Considering the rider as a projectile, you can use the kinematic equations to describe their vertical and horizontal motion. For vertical motion, the equation
`\(y(t) = y_0 + v_(0y)t - (1)/(2)gt^2\)` can be used, where
`\(y_0\)` is the initial height,
`\(v_(0y)\)` is the vertical component of the initial velocity, (t) is time, and (g) is the acceleration due to gravity. For horizontal motion, the equation
`\(x(t) = x_0 + v_(0x)t\)` can be used, where
`\(x_0\)` is the initial horizontal position, and
`\(v_(0x)\)` is the horizontal component of the initial velocity.

Step-by-step explanation:

In vertical motion, the equation
`\(y(t) = y_0 + v_(0y)t - (1)/(2)gt^2\)` describes the snowboarder's height at any given time during the jump. Here,
`\(y_0\)` is the initial height,
`\(v_(0y)\)` is the vertical component of the initial velocity, (t) is time, and (g) is the acceleration due to gravity
`(\(9.8 m/s^2\))`. This equation takes into account the initial height, the initial upward velocity, and the effect of gravity on the snowboarder's height.

For horizontal motion, the equation
`\(x(t) = x_0 + v_(0x)t\)` describes the snowboarder's horizontal position at any time during the jump. Here,
`\(x_0\)` is the initial horizontal position,
`\(v_(0x)\)` is the horizontal component of the initial velocity, and (t) is time. This equation assumes constant horizontal velocity, neglecting air resistance.

By solving and analyzing these equations, you can gain insights into the snowboarder's trajectory during a jump. Numerical methods like Euler or Runge-Kutta can be applied to simulate the motion, allowing you to investigate the feasibility of a quintuple cork under different initial conditions for the jump and rider.