Rock Climbing and Differential Equations: The Fall-Factor

Dr. Dan Curtis

Central 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.

• 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.

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

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

2

2

d x dvv

dt dx

dvv gdx

21

2v gx C

During free-fall

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 mL

v D gD

2 22 ( )Fk

v gx x DmL

The solution is

Maximum force felt by the climber occurs when

Tx D and 0v

20 2 ( )T T F

kgD D D

mL

max 2 TDF mgkL

2 2( )T F T TD D k mgLD mgkDk

L L k L

The maximum force is given by