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Rock Climbing and Differential Equations: The Fall-Factor Dr. Dan Curtis Central Washington University 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. Climbers use ropes and protection devices placed in the rock in order to minimize the consequences of a fall. • 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. protection point climber belayer protection point climber belayer protection point climber belayer L = un-stretched length of rope between climber and belayer. DF DT 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. Fall-factor about 2/3 Fall-factor 2 belay point position at start of fall 0 DF position at end of free-fall DT position at end of fall x During free-fall 2 d x m 2 mg dt 2 d x dv v 2 dt dx dv v g dx 1 2 v gx C 2 v0 x0 when so C 0 v 2 gx 2 When x DF vF 2 gDF After the rope becomes taut, the differential equation changes, since the rope is now exerting a force. dv k v g x DF dx mL v( DF ) 2 gDF The solution is k 2 v 2 gx ( x DF ) mL 2 Maximum force felt by the climber occurs when x DT and v0 k 2 0 2 gDT ( DT DF ) mL The maximum force is given by DT DF k 2mgLDT 2mgkDT k( ) L L k L Fmax DT 2mgk L