Rock Climbing and Differential Equations: The Fall-Factor

1. Rock Climbing and Differential Equations: The Fall-Factor Dr. Dan Curtis Central Washington University

2. Based on my article: “Taking a Whipper : The Fall-Factor Concept in Rock-Climbing” The College Mathematics Journal, v.36, no.2, March, 2005, pp. 135-140.

3. Climbers use ropes and protection devices placed in the rock in order to minimize the consequences of a fall.

7. Intuition says: The force exerted on the climber by the rope to stop a long fall would be greater than for a short fall. • According to the lore of climbing, this need not be so.

8. protection point climber belayer

9. protection point climber belayer

10. protection point climber belayer

11. L = un-stretched length of rope between climber and belayer.

12. DF DT

13. The Fall-Factor: DT / L Climbing folklore says: The maximum force exerted by the rope on the climber is not a function of the distance fallen, but rather, depends on the fall-factor.

14. Fall-factor about 2/3

15. Fall-factor 2 belay point

16. position at start of fall 0 position at end of free-fall DF position at end of fall DT x

17. During free-fall

18. when so When After the rope becomes taut, the differential equation changes, since the rope is now exerting a force.

19. The solution is

20. Maximum force felt by the climber occurs when and

21. The maximum force is given by