1. Modeling a Safe Ski Jump Landing
Andrew Swedberg
MA3232

2. 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

3. Existing Terrain Park Jumps

4. 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

5. 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

6. 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)

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

8. 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= 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, radians)

9. 5 m Landing Surface 10 degree jump
Velocities of 20 mph and higher produce a poor design. 10 and 15 mph are good designs.

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

11. 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!

12. 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

13. 10 m Landing Surface 20 degree jump
Making the takeoff steeper produces a poorer design at higher velocities

14. 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. 15 m Landing Surface 10 degree jump
By increasing the jump landing surface to 15 meters, most all velocities would be adequate

16. 15 m Landing Surface 20 degree jump
Making the takeoff steeper produces a poorer design at higher velocities

17. 15 m Landing Surface 30 degree jump
Again, making the takeoff steeper produces a poorer design at higher velocities

18. 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

19. 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.