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

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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.

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

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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.

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belayer climber protection point

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belayer climber protection point

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belayer climber protection point

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

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DFDF DTDT

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The Fall-Factor: D T / 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.

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Fall-factor about 2/3

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Fall-factor 2 belay point

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0 x DFDF DTDT position at start of fall position at end of free-fall position at end of fall

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During free-fall

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

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The solution is

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Maximum force felt by the climber occurs when and

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The maximum force is given by

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