Rock Climbing and Differential Equations: The Fall-Factor
Dr. Dan CurtisCentral Washington University
Based on my article:
“Taking a Whipper : The Fall-FactorConcept 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.
• 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
climberprotection point
belayer
climberprotection point
belayer
climberprotection point
L = un-stretched length of rope between climber and belayer.
DF
DT
The Fall-Factor is defined as the ratio
DT / L
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.
Fall-factor about 2/3
Fall-factor 2
belay point
0
x
DF
DT
position at start of fall
position at end of free-fall
position at end of fall
2
2
d xm mgdt
During free-fall
2
2
d xm mgdt
2
2
d x dvvdt dx
During free-fall
2
2
d xm mgdt
2
2
d x dvvdt dx
dvv gdx
During free-fall
2
2
d xm mgdt
2
2
d x dvvdt dx
dvv gdx
212v gx C
During free-fall
0v when 0x so 0C
0v when 0x so 0C 2 2v gx
0v when 0x so 0C 2 2v gx
Fx DWhen 2F Fv gD
0v when 0x so 0C 2 2v gx
Fx DWhen 2F Fv gD
After the rope becomes taut, the differential equation changes, since the rope is now exerting a force.
( ) 2
F
F F
dv kv g x Ddx mLv D gD
( ) 2
F
F F
dv kv g x Ddx mLv D gD
2 22 ( )Fkv gx x DmL
The solution is
Maximum force felt by the climber occurs when
Tx D and 0v
Maximum force felt by the climber occurs when
Tx D and 0v
20 2 ( )T T FkgD D DmL
max 2 TDF mgkL
2 2( )T F T TD D k mgLD mgkDkL L k L
The maximum force is given by