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

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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.

belayer climber protection point

belayer climber protection point

belayer climber protection point

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

DFDF DTDT

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.

Fall-factor 2 belay point

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

During free-fall

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

The solution is

Maximum force felt by the climber occurs when and

The maximum force is given by

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