Physics and Falling Rock Climbers
                                      By Matt Covington

                     As a rock climber, and a physics student, I have begun to wonder exactly how far I can fall and
                not go over the breaking strength of my rope.  This page is dedicated to the exploration of that
                problem.  The problem will be solved using the impulse equation and basic knowledge of climbing.

 We start with the equation:

                       F = DP / DT                                     DKE = -DPE                       F = force on rope due to fall
                                                                                                                             P = momentum of climber at
                       F = P - P₀ / DT     P=0                     1/2 mv² = -mg (y - y₀)                 point of deceleration (POD)
                                                                                                                             T = time in seconds
                       F = -P₀ / DT          P=mv                   v² = -2g( y - y₀)                    m = mass of climber
                                                                                                                             v = velocity of climber at POD
                       F = -mv / DT                                     v = sqrt(-2g(y - y₀))              g = acceleration due to gravity
                                                                                                                             y = height of climber after fall
                       DT = DD /  DV_avg  (assuming and accel is constant)                             (zero for convenience)
                                                                                                                             y₀ = height of climber before fall
                       DV_avg = v / 2                                                                                D = distance required to stop climber
                                                                                                                            S_f = stretch factor of rope
                                                                                                                                   (converted to decimal)
                      DT = 2DD / v                                                                                 L = length of rope from climber to belayer
 
                      F = -mv² / 2DD          v = sqrt(-2g(y-y₀))
                                                        v² = -2g(y - y₀)

                      F = 2mg (y - y₀) / 2DD    y=0
 
                      F = -mgy₀ / DD        In a climbing fall, once the deceleration has begun, (DD = S_f * L)
                                                      DD = rope stretch factor times the length of the rope from the climber to the
                                                      belayer (as shown in figure).  Stretch factor is given in percentages so it
                                                      must be converted from percentage to decimal form.

                     F = -mgy₀ /  (S_f * L)      Is our final falling equation.

                Before fall                                  After fall
 

Now lets graph a typical fall situation.  We'll use a climber that weighs 65kg, a rope with 2% stretch
and 20 m length between the climber and belayer.

m = 65 kg
g = 9.8 m/s/s
S_f  = .02
L = 20 m
x = distance of fall

So:  F = -(65)(9.8)(x) / (.02 * 20)
 
       F = -1592.5x
 
 

 Graph of above equation.

 

      Ok, now lets discover the limits of these rope things.

      If a 75 kg climber was climbing on 2% stretch rope that could hold 22250 N (5000 Lbs),
how far could he fall without breaking the rope, when he has 15m of rope between himself and
the belayer..

m = 75 kg
F = -22250 (or slightly less than)
S_f  = .02
L = 15 m
x = max distance of fall

F = -mgx / (S_f * L)

-22250 = -(75)(9.8)(x) / (.02 * 15)

-22250 = -735x / .3

-22250 = -2450x

x = 9.08 m

     So, the climber could fall about 9 meters without breaking the rope.  However, at this speed the
climber wouldn't be worried about the rope breaking.  He would be worried about the force of the
deceleration snapping him in half.  The strength of the human body usually becomes the limiting factor
in climbing falls, not the strength of the rope.  Now we see why.  Human skeletons usually can't
withstand 20, 000 Newtons of force, climbing ropes usually can.
 

     Below are figures of the two types of climbing, lead and top rope.  In these different types of climbing
the climber is exposed to different fall possibilities.  In lead climbing, the climber falls twice the distance from
himself to the anchor.  In top roping the climber falls only far enough to take the slack out of the rope, which
is usually no more than a few feet.  This is why there are two different types of rope for climbing.  One is
called Dynamic.  It is used for lead climbing and has a high stretch factor.  You can see in the equation why
this would help reduce the force of the fall.  The other kind of rope is called static and is used for top roping
and rapelling.  It's stretch factor is much lower and  it cannot absorb as much shock.  This is why it is only
used for situations where you will not be falling very far or if you need rope that isn't so stretchy.

 Lead Climbing                            Top Roping

Here are some links on similar topics:
 
 Physics and Momentum
 www.science.urich.edu/~rubin/pedagogy/131/131notes/131notes_118.html
 www.cord.edu/dept/physics/p128/lecture19.html#topic1
 www-astro.physics.uiowa.edu/~rlm/physics1/labs/Labs/lab9.html
 www.physics.harvard.edu/courses/1996_97/phys15c/lecture14/lc14w.css

Ropes and Climbing
 Static Rope
 Dynamic Rope
 Climbing basics
 Cool climbing site
 

     I constructed this web page for my AP Physics class at Fayetteville High School.  I chose this topic
because of my love for rock climbing and curiousity about the physics involved.  If you have any questions
or comments please send them to my email address at  MCovington@Arkansasusa.com