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
v0
x0
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
v0
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 