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Thread: Guide: How to Find Proximate Height of a Cliff

  1. #1
    Moderator jman's Avatar
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    Guide: How to Find Proximate Height of a Cliff

    Moab mark has inspired to me to become more technical on the boards here, and to start I thought I would post this:

    Alright, I AM most sure most of you know this, but I had a talk with one of my friends who is an expert canyoneer but he didn't know a good way to measure distances of a cliff - in scenarios where he had to escape the canyon proper and possibly rappel, but did not want to throw the rope down just to see if it was long enough...

    Thank goodness for math:

    Here is the formula (and I personally think this should be taught part of a CORE theorem of Canyoneering):

    H=1/2GT^2

    H=Height
    G=Gravity (9.8 m/sec)
    T=Time
    ^2=squared

    Now practical use:
    Drop a rock off the edge of a shear cliff.
    1) As soon as you drop it, you have to time it exactly.
    2) As soon as it hits, stop your timer.
    3) Plug in the values into the formula.
    4) H=1/2(9.8)(1.8)^2

    So the above for the layman is saying, H= 1/2 * 9.8 (gravity velocity) * 1.8 (seconds)^2.

    So 1.8^2 = 3.24 (seconds).

    Thus, H=1/2(9.8)(3.24) = 15.876 m (remember cause that's what base of gravity is. Technically, you could convert 9.8m into feet, which would be 32.15 (rounded), and then you could avoid the next step)

    So, you can be satisfied with the meter base, or you can convert it to feet, which equals....
    15.876 m
    (1 meter =3.28 feet (rounded)
    = 52.07 feet.

    Now, the formula above does not take into account for air resistance, and that's why the formula should be taken as a "rough guess" and not EXACT. Just a warning.

    I have tested this even on places where I had to lob a rock to make it over the edge, and with that, I started my timer (on my watch) exactly when it started to descend when it reached the same height as I was, and when I heard it hit, I added in another 25+ feet just for safe measure. After the rappel, I learned that I had about 22 feet extra, so I was ONLY off 3 feet!. So, the 25 feet was a good added safety measure no doubt, but overall, the method tends to be very reasonably accurate.

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  3. #2
    Very cool - thanks jman. I look forward to trying this!

  4. #3
    Quote Originally Posted by Cirrus2000
    Very cool - thanks jman. I look forward to trying this!

    So a second is close to 30'.
    Mark

  5. #4
    Jman,

    You forgot one very important step..........Yell Rock before tossing.

  6. #5
    Bogley BigShot oldno7's Avatar
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    Re: Guide: How to Find Proximate Height of a Cliff

    Quote Originally Posted by jman

    Here is the formula (and I personally think this should be taught part of a CORE theorem of Canyoneering):

    H=1/2GT^2

    H=Height
    G=Gravity (9.8 m/sec)
    T=Time
    ^2=squared
    I always keep this formula in my top right shirt pocket, in my vinyl pencil holder. My slide ruler is alway's attached to the shoulder strap of my pack, with 2mm accessory cord. This is really old stuff jman.......everyone does it......don't they?

  7. #6
    very cool thanks for the info., cant wait to try it out.

  8. #7
    A few years back we were exploring new canyons around Ouray, and Cascade Creek caught our eye. This canyon ends with a nice waterfall at the edge of town that is a very popular and short hike in. We heard conflicting reports from ice climbers how tall this waterfall is - some said 200', some even some less, some hinted at more. The always resourceful John Hart brought a laser range finder one day and measured it from the bottom to a potential tree anchor at the top as around 280', and he was almost exactly right.


    M

  9. #8
    Very cool!

    One factor I'm curious about. since the rock is going from a complete stop to it's maximun speed of 9.8 m/sec. How fast is the accelleration of the rock? Does it achieve maximun speed in .5 seconds, 1 second, 2 seconds, 3 seconds?
    Your safety is not my responsibility.

  10. #9
    reedus started a thread on this in 2006

    http://bogley.com/forum/viewtopic.php?t=1982

  11. #10
    the 9.8 is the acceleration due to gravity, not max speed... so this method is a rough guess. after 1 second, the rock is traveling at 9.8 m/s, at 2 seconds, it is traveling at 19.6 m/s, etc. of course, this isnt counting if it bounces of the canyon wall, if it isnt falling down a straight cliff, etc. and not counting wind resistance as mentioned, although that would likely be minimal in this case....
    "Human beings were not meant to sit in little cubicles staring at computer screens all day, filling out useless forms and listening to eight different bosses drone on about mission statements"

    Peter Gibbons - Office Space

  12. #11
    What is this "Meters per Second" of which you speak?!?

    I remember the first week of high school physics... back when dirt was new.... they had us dropping marbles off everything on the school grounds and calculating a height.... if your timing is good it's amazingly accurate....

    Probably never get to do this at school now days.... just imagine the fear falling objects would inflect upon kids who are too candyass to play dodge ball.


  13. #12
    Content Provider Emeritus ratagonia's Avatar
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    Quote Originally Posted by Deathcricket
    Very cool!

    One factor I'm curious about. since the rock is going from a complete stop to it's maximun speed of 9.8 m/sec. How fast is the accelleration of the rock? Does it achieve maximum speed in .5 seconds, 1 second, 2 seconds, 3 seconds?
    What, they don't teach physics in Utah schools?

    Jman - the accuracy problem is in measuring the time interval. For distances less than 600 feet, the effects of the air resistance are very small. To measure accurately, use your watch to get it as precise as possible, and do it three times and average the times. If having to toss it upward, start timing when it reaches its peak, rather than when it passes you. Throwing it outward has no effect on how long it takes to fall downward, but it may hit further out, which is often further down.

    (Sidenote: how about a bullet? A bullet starts falling as soon as it leaves the muzzle, but its 'fall' is reduced by the lift generated by its shape and speed through the air. That's the old monkey-and-the-hunter problem...)

    Cricket-O-Death: gravity causes the objects to accelerate, at least until they hit the ground. The acceleration is: (9.8 meters per second (a speed)) per second. So, after one second, the rock is going 9.8 m/s. After two seconds, the rock is going 19.6 m/s. After 3 seconds, 29.4 m/s. In American units, it is 32 feet per second, per second.

    Air resistance slows the acceleration, and eventually balances with the force of gravity, at which point we say it has reached "terminal velocity". The terminal velocity of any object depends on its weight and its air-resistance-profile, but a rock big enough to hear hit will not reach its terminal velocity in 600 feet.

    [quote=the wiki]
    Based on wind resistance, for example, the terminal velocity of a skydiver in a free-fall position with a semi-closed parachute is about 195 km/h (120 mph or 55 m/s). This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the body balance each other more and more closely as the terminal velocity is approached. In this example, a speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on. Higher speeds can be attained if the skydiver pulls in his limbs (see also freeflying). In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s), which is also the terminal velocity of the peregrine falcon diving down on its prey. And the same terminal velocity is reached for a typical 150 grain bullet travelling in the downward vertical direction

  14. #13
    Quote Originally Posted by DWayne27
    the 9.8 is the acceleration due to gravity, not max speed... so this method is a rough guess. after 1 second, the rock is traveling at 9.8 m/s, at 2 seconds, it is traveling at 19.6 m/s, etc. of course, this isnt counting if it bounces of the canyon wall, if it isnt falling down a straight cliff, etc. and not counting wind resistance as mentioned, although that would likely be minimal in this case....
    Thank you. that was the part I was missing. I always thought a rock would quickly accellerate to a certain speed determined by the force of gravity pulling on it (minus wind resistance), and stay that constant speed once achieved. Yeah, physics aren't something I understand well.




    Edit: Thanks for add'l info Tom! Didn't see when reading initial response from DWayne. FYI: I went to school in liberal San Diego, I blame them! Too busy trying to catch up the non english speaking students. And of course that was 15 years ago so even though they probably taught me correct, it's long since vacated my brain since it's something I never use. :)
    Your safety is not my responsibility.

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    Moderator jman's Avatar
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    I'm impressed, you guys know your physics. I didn't want to get too techinical with gravity and reaching terminal velocity....unless u really want to. That's why I said it was a "rough guess".

    Tom, u are correct, the timing is crucial. I should of emphasized that more.

    Moab, for a rough estimate 1 sec is close to 30, but after 1 sec, gravity is making it accelerate till it reaches terminal velocity, so 3 does not equal 90 feet.

    Dwanye, I was going to initially put that in such as other factors to avoid, eg. not letting the rock hit the canyon walls, not to use something that "floats" etc, but figured that was practical sense and left it out.

    Stefan, I searched for "finding height" and no results showed up.

    Oldno, I assumed most would know this, but canyoneering topics are slow these days and thought some could benefit. I do the same and carry a very small notepad and pencil in my pack and have a backup in my "possibles bag" too.

  16. #15
    Content Provider Emeritus ratagonia's Avatar
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    Quote Originally Posted by Deathcricket
    Thank you. that was the part I was missing. I always thought a rock would quickly accellerate to a certain speed determined by the force of gravity pulling on it (minus wind resistance), and stay that constant speed once achieved. Yeah, physics aren't something I understand well.
    Uh huh, that's called Aristotlian Physics, a good first approximation. Persisted for quite a long time, until two new guys came along: Galileo Galilei and Isaac Newton. Now we have Newtonian physics for not-near-light-speed situations.

    Good to see lots of people jumped on this one. HEY! Don't you people have jobs???

    Tom

  17. #16
    Content Provider Emeritus ratagonia's Avatar
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    Quote Originally Posted by moab mark
    Quote Originally Posted by Cirrus2000
    Very cool - thanks jman. I look forward to trying this!

    So a second is close to 30'.
    Mark
    Actually, AFTER one second, it is falling at 32 feet per second.

    DURING THE FIRST SECOND, it would fall half that, or 16 feet.

    Tom

  18. #17
    Quote Originally Posted by ratagonia
    Actually, AFTER one second, it is falling at 32 feet per second.

    DURING THE FIRST SECOND, it would fall half that, or 16 feet.
    You don't even need to remember the formula or do any real math if you just remember that.....


  19. #18
    Content Provider Emeritus ratagonia's Avatar
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    Quote Originally Posted by Iceaxe
    Quote Originally Posted by ratagonia
    Actually, AFTER one second, it is falling at 32 feet per second.

    DURING THE FIRST SECOND, it would fall half that, or 16 feet.
    You don't even need to remember the formula or do any real math if you just remember that.....

    Here's the chart for you Shane, and for any other math-o-phobic engineers out there:
    Attached Images Attached Images  

  20. #19
    Quote Originally Posted by ratagonia
    Here's the chart for you Shane, and for any other math-o-phobic engineers out there:
    You have to admit.... it's easier to carry the chart than it is your slide ruler....

    And you know us engineers... it's all about practicality.

    .

  21. #20
    Moderator jman's Avatar
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    Now, I can't easily impress my friends now with my l33t math skills, with that chart. Not nearly as impressive. haha

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