Modeling a Safe Ski Jump Landing Andrew Swedberg MA3232

Outline Existing Terrain Park Jumps Background of the Jump Setup for the Problem ODE used to design the land surface Variation of parameters Nine example versions My recommendation

Existing Terrain Park Jumps

Background of the Problem Desire to reduce ski and snowboard injuries from poor landings on terrain park jumps Current ski jumps are designed by skiers/snowboarders based on experience alone (no mathematical analysis to reduce risks and enhance safety) Initial research and article produced by Professor Mont Hubbard, UC Davis

Setup of the Problem GOAL: Find the optimum combination of the parameters to produce a safe and reasonable ski jump landing slope. Lower perpendicular velocity = lighter impact upon landing = less risk for injury The optimal design is when the landing slope is nearly parallel to the skier’s path upon landing

Landing Surface Design First Order Differential Equation with multiple parameters ys: shape of landing surface Θ0: Takeoff Angle: Angle of jump and skier takeoff, in radians Θ: Angle of jump that the skier jumps with. In this case, assumed to be the same as Θ0 g: gravity: 9.8 m/s h: Equivalent Fall Height: Comfortable height in which skier may fall v0 : Skier Initial Velocity x: horizontal distance (independent variable)

Steps to Produce the ODE Θ0=β0+ΔΘ In this model, Θ=Θ0 Skier simply glides off of the jump.

Variation of Parameters x: horizontal distance of the jump (5, 10, 15) v0: Skier Initial Velocity: from 10 mph to 30 mph in 5 mph increments In meters/second: (4.47, 6.71, 8.94, 11.76, 13.41) 1 mph= 0.447 m/s h: Equivalent Fall Height (EFH): vertical height of comfortable falling distance: 1 m Θ0 : Angle of jump and skier takeoff, in radians 1 degree = pi/180 radians (10, 20 ,30 degrees) (0.175, 0.349, 0.523 radians)

5 m Landing Surface 10 degree jump Velocities of 20 mph and higher produce a poor design. 10 and 15 mph are good designs.

5 m Landing Surface 20 degree jump Velocities of 20 mph and higher produce a poor design

5 m Landing Surface 30 degree jump Big difference in design at 10 and 15 mph vs. the higher velocities 20, 25, and 30 mph produce a very poor (opposite slope) design!

10 m Landing Surface 10 degree jump By allowing the landing slope to increase to 10m, there are more options to build a good model. 25 and 30 mph are still too fast

10 m Landing Surface 20 degree jump Making the takeoff steeper produces a poorer design at higher velocities

10 m Landing Surface 30 degree jump With the very steepest jump, a skier would have to be going relatively slow to land on a good sloped landing

15 m Landing Surface 10 degree jump By increasing the jump landing surface to 15 meters, most all velocities would be adequate

15 m Landing Surface 20 degree jump Making the takeoff steeper produces a poorer design at higher velocities

15 m Landing Surface 30 degree jump Again, making the takeoff steeper produces a poorer design at higher velocities

Recommendation: 10 m Landing Surface 10 degree jump 15 mph This model produces a safe design that does not require a lot of snow to build: about 9m tall by 10 m long

Conlusion I intend on pursuing this in greater detail Goal is to have a thesis for an MS in mathematics, while providing a tangible product to ski area in order to ensure safety for all skiers and snowboarders.