Rock Climbing and Differential Equations: The FallFactor

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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 Fa
ll-Factor Concept in Rock-Climbing The Colleg
e Mathematics Journal, v.36, no.2, March, 2005, p
p. 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.
  

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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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protection point
climber
belayer
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protection point
climber
belayer
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protection point
climber
belayer
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L un-stretched length of rope
between climber and belayer.
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DF
DT
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The Fall-Factor is defined as the ratio DT / L

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The Fall-Factor is defined as the ratio DT / L
Climbing folklore says
The maximum force exerted by the rope o
n the climber is not a function of the dist
ance fallen, but rather, depends on the fal
l-factor.
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Fall-factor about 2/3
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Fall-factor 2
belay point
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position at start of fall
0
position at end of free-fall
DF
position at end of fall
DT
x
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During free-fall
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During free-fall
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During free-fall
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During free-fall
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when
so
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when
so
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when
so
When
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when
so
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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Maximum force felt by the climber occurs when
and
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The maximum force is given by