UKC

My friend fell free from a 20 foot runout before the rope stopped him. Any idea what his maximum falling speed was?

Here you go:

https://www.angio.net/personal/climb/speed

Secondary school physics/applied maths.

v^2 = u^2 + 2as

u=0

a=9.8m/s^2

s=20ft=6.1m

Thus, v=10.9m/s

Thus, v=7.73m/s

Not sure that is right for two reasons:

1. It's a 20' (6.1m) run out, so in theory, the actual distance is 40' (12.2m).
2. Your calculation if applied to 20' (6.1m) seems to negate the multiplier factor of 2 in the '2as' part of the equation. You git, you've edited your post to hide your mistake...

what about air resistance? Is this significant? Both above answers assume it out. Also the calculator method asks your weight - why?

Over that distance, air resistance would be minimal.  Friction of the rope running through the gear might have an effect though. I assume friction applied by the belay device can be ignored as the full height of the fall would have been achieved before belaying could be applied (you could actually take in slack during the fall and this might decrease final velocity.  However, it would increase the Fall Factor).
The weight inclusion is an extra (superfluous) bit of data that allows the energy to be calculated.

Are you wanting to be able to calculate that on the fly?

So for instance if youre 8m above your last piece and think you might fall off, you would work out that you'll reach terminal velocity and think ah f*ck that ill down climb instead.

However if you were only going to max out at 14.21m/s you would just go for it?

Yes, I forgot the factor 2, then double checked, and edited the post...

ps. You edited your post, too, 2 minutes after you posted. The same time it took me to edit mine...

You've got a point about the runout, though... 

Because it calculates things other than maximum velocity, which require mass.

My edit was purely down to correcting a grammatical error.  Yours was down to you being a fool that rushed in... 

I should've turned up to school more... 

Thanks everyone.

Next question. Under these circumstances, how long are you in free fall? that is, until the belayer begins to catch you. Assuming he doesn’t take in during your fall.

When I saw the headline, I thought this was going to be another thread about Corbyn or May

If you took in slack doesn't less rope out decrease the fall factor, as the distance fallen might now be slightly reduced?

..assuming the climber is above a few bits of gear and not doing a FF2...

Eg, 4m above gear at 8m, fall 8m on 12m rope = 0.666

Pull 1m slack in as falling, fall 7m on 11m rope, = 0.636

Or fall 8m from  4m above 1st gear on a hanging belay sport route so u don't deck = 8/8, but yard in 1m slack and its then 7/7 ? So both ff1 if from above 1 piece gear.?

If its a fall factor 2 scenario, ff=2 , eg fall from 4m above belay= fall 8m on 4m rope, 8/4 =2, if no slack taken in, but if 1m slack taken in, ff increases, yes.

Eg fall now 7m on 3m rope = 2.333

Climb 6m above 1st gear at 4m, =10m rope out, fall off (multi pitch, hanging belay), fall 6m to gear + 6m past it = 12m, on 10m rope = ff 12/10 = 1.2.

Pull in 1m slack as falling, fall 6m to gear + 5m past it = 11m on 11m of rope, ff = 1.

Where do you get 11m of rope out in the second case? Surely it should be the original 10 reduced by one, so 9. FF then INCREASES slightly to 1.222

You can do this one yourself as a an exercise

S = UT + 1/2AT^2

Solve for T

When I'm doing it in my head I generally regard A =~ 10 as plenty good enough.

Cheers John, think yr right, in a rush to get out + do some 4 real now, will double check later... Interesting how it differs at various points on a route!

Might b other errors too...!

Should reach terminal velocity if you push them off a tall enough building. The real question is who would hit he ground first.

Lots of Dislikes.
Think I should have added the to indicate that I was obviously pulling the captains leg. 

In reply to captain paranoia:

Read my post above yours.
Apologies if it came across differently.

This two minutes thing is becoming a habit. You hadn't replied when I started composing...

I'm trying to promote you to major paranoia...

One of them is so full of hot air that they'd never hit the ground.

Terminal velocity of a falling person is usually stated as just above 50 m/s reached after just over 10 seconds.  If you do the calculation assuming no air resistance after 1 second you fall about 5m and reach about 10m/s, after 2 seconds you fall about 20m and reach about 20m/s and aftter 4 seconds you fall about 80m (getting close close to the biggest falls likely on a climbing belay)  reaching about 40m/s.

I thought the 50m/s was for a skydiver in that starfish position they use to maximise air resistance & control? A falling climber is likely to be more or less upright and far more streamlined.

Are you seriously quibbling on such a point when you need to fall more than 20m to get any significant air resistance?

To help show there is not that much difference in body position (20-40%); hardly unexpected as resistance force increases significantly with velocity (vs gravitational force remaining constant).

https://en.m.wikipedia.org/wiki/Speed_skydiving

i never got fall factor.  It seems to me, perhaps naively, that the only important thing  is how fast you are going when you reach the end of the drop.

The clothing on a body falling at terminal velocity makes a very distinctive rattling sound as the body goes past you. It's quite alarming. We found this out when we were quite low down on a route in the Verdon, and had base jumpers falling past us, and opening their chutes below us. 

It's not the fall that does the damage, it's the sudden stop. Longer falls have a correspondingly lengthier piece of rope to absorb the impact. When you do the maths, the thing that determines the severity of the stop is the ratio of distance fallen over rope length.

My physics is very poor. In a longer fall of given factor isn't the impact force the same but spread over a long period?....If this so it could actually cause more damage to spine etc?

No, it's how quickly you stop that does the damage; what acceleration you (and the rope and gear) are subjected to as you stop. The more rope you have out, the more rope there is to stretch, so the lower the force, even if that force acts for longer. It's high acceleration that breaks/rips gear and spines.

Thats not strictly true. In theory, a higher fall factor fall with more rope out will have higher force than a shorter fall with lower fall factor but less rope out . That is to say the braking force should in theory be roughly the same for the same fall factor whatever the length of fall, as the rope is elastic. A longer fall with the same fall factor has the same force applied longer to absorb the extra energy. Friction around runners, slippage on the belay device and 'give' in the belay system further reduce the force.

The force in a FF2 is pretty brutal and can do a falling body (and the belayer) some serious damage. A rope should hold but a sling will snap with that force. People need to understand fall factors in this sense so they don't do stupid things on multipitch trad like tie their lead rope  into  the first belay point when they arrive at a belay via a sling then climb above it to another runner. Clipping the sling direct to the harness is just as risky. If you fall at the higher belay point it could be fatal. It is important to treat the first belay point as a standard runner if you are climbing up a bit more for another belay point. It may well feel less secure for those who don't understand the forces involved but is much much safer.

Yes, I was being rather sloppy about rope out.

I've always found this analysis to be very useful; even if you can't follow the mechanics, the summary findings the end are useful.

http://lamountaineers.org/pdf/xRopes.pdf

No I wasn't "quibbling". I didn't have the slightest idea whether the difference was 20-40% or 300%. That's why  asked the perfectly genuine (if slightly off topic) question.

I was under the impression asking genuine questions in order to learn things was generally regarded as a reasonable sort of thing to do.

Just as a matter of interest, did the leader fall, or did you push him?

I can see great opportunities for the stats obsessed commentators at the Olympics here. 

'And Ivan Soxxoff reached a record terminal velocity of 16.5 m/s in that fall, that puts him in the lead of the downward bound trophy'

How daft, that only works in the real world. In the world of online phorums, every question is a sign of agression and every remark a deglaration of war. 

Almost every time a falling climber reaches 0mph, 0m/sec. Hopefully the rope causes the stop and it is gentle. Sometimes the ground will determine the final speed quite suddenly!

My longest fall high up on the Etive Slabs.

None of the physics quoted on here seems to apply.....

I took an etive kiss in an unexpected snow flurry about 20 years ago. 6m fall at most and very much in slow motion due to the friction  but I can still feel the soreness in my elbow and remember the loss of skin.

So taking slack in can increase ff on long falls above low gear or no gear, which is maybe significant? increasing in some cases from ff2 to 2.3,

but ff decreases, and not very significantly, if taking slack in on smaller falls high up above several pieces of gear, eg 0.66 down to 0.63.

Should / will anyone remember when / how to apply this?

But the overiding factor that decides weather or not it might be handy to take slack in is to protect the leader from hitting an obstacle or the ground....

12m fall (40ft) gives a velocity of roughly 15.5m/s (34mph) not accounting for drag. Drag at 35mph on a human body falling feet first is minimal by comparison with their weight so no, drag doesn't make much difference.

Aerodynamic drag equation is: F_drag = 1/2 * rho * v^2 * Cd * A https://en.wikipedia.org/wiki/Drag_equation

Plug in some rough numbers: assume the climber is a long cylinder https://en.wikipedia.org/wiki/Drag_coefficient assume they're as fat as me and at a reasonable surface air density of 1.2 :

0.5 * 1.2 * 15.5 * 15.5 * 0.82 * 0.1 = 11N

Probably an underestimate since real climbers are knobbly and flap but an irrelevant 1% of my weight at 9.8 * 88 = 862N

jk

Work it backwards....drag force equals gravitational force, say at around 60m/s

In theory at least, the fall factor could be infinite if you take slack in mid-fall. Imagine a fall from 1m where the belayer takes in 1m of slack.

It makes sense on reflection: Taking in during a tiny fall high up could reduce the fall distance a lot (e.g. a 2m fall could end up only 1m) while having a much smaller effect on the amount of rope out.

The turning point is when the fall factor is exactly 1, i.e. the runout is identical to the height of the top gear. In this case, taking in has no effect at all on the fall factor. More severe falls still will see a rise in fall factor by taking in.

Obviously ignoring stretch is a big simplification but the principle should remain pretty valid.

The record for headfirst skydiving seems to be about 166m/s (601kph) which gives a CdA of 0.053 assuming 90kilo with parachute. That gives a Cd of roughly 0.53, slicker than expected but it is a world record.

Aero' drag is essentially irrelevant in short falls.

jk

Wasn't that record at altitude?

Ten dislikes for providing an explanation using the appropriate applied mathematics...?

Don't worry, I got 15 for making a joke about your pre-edited mistake!
That's UKC for you...

Probably, there's footage and the landing zone looks alpine. It's not the giant weather balloon jump by Felix Baumgartner.

jk

How far does a leader have to fall to give him time to calculate what speed he is going?

African or European?

I think it was your assumption of 20ft instead of 40ft.

Not convinced it was worth dislikes though