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.

8. 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.

9. protection point climber belayer

10. protection point climber belayer

11. protection point climber belayer

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

13. DF DT

14. The Fall-Factor is defined as the ratio DT / L

15. The Fall-Factor is defined as the ratio 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.

16. Fall-factor about 2/3

17. Fall-factor 2 belay point

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

19. During free-fall

20. During free-fall

21. During free-fall

22. During free-fall

23. when so

24. when so

25. when so When

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

28. The solution is

29. Maximum force felt by the climber occurs when and

30. Maximum force felt by the climber occurs when and

31. The maximum force is given by