Altitude Sickness

Act 1

View from Pikes Peak:
On a trip out west with my wife, we took a trip up to the top of Pikes Peak by Colorado Springs. I’m calling this post Altitude Sickness because despite having this post in mind when we went up, I didn’t take the one picture that I wanted to take.  Idiot. Nincompoop. Sigh. Lets move on. Here is a very similar picture (from wikipedia) to the one I wanted to take:

At the top of Pikes Peak I sealed a very similar water bottle. Pikes Peak is 14,110 ft tall. I took the water bottle back home and here is a picture of the water bottle.

So. What altitude am I at right now?

Act 2

You’ll need the following bits of information to solve.

  • P_{1}V_{1}=P_{2}V_{2}
    Where P is pressure, and V is volume.
  • P=P_{0}(1-2.25577*10^{-5}*h)^{5.25588}
    P is pressure, P_{0} is the pressure at sea level (101,325 pascals), and h is the altitude in meters.
  • The volume of the crushed (by air) bottle is 395 mL (using Archimedes’s water displacement method and the fact that 1mL of water = 1g of water. Video of measurement found here.)
  • The volume of the bottle is 555 mL.

Act 3

The pressure-altitude formula is used twice. Once to get the air pressure at the top of Pikes Peak. This information can then be used to find the air pressure for my current location. Then you need to solve for h in the pressure-altitude formula to find the current altitude. The answer ended up being 162m, while the internets puts me at 91.4m of altitude. Not too shabby! Pretty damned good if you ask me.


  • Take the right picture at the top of Pikes Peak.
  • Keep the bottle crushed (I had to open it up back at home to get the volume of an uncrushed bottle).
  • Get an identical bottle for comparision.
  • … (any more)


  1. Is the question: What altitude am I at right now” legit? Do you think this question is natural to the students? I’m sure there’d have to be some explanation given to why the bottle gets crushed. But will this question come up? Give me your best read.
  2. Should I keep the height of the mountain hidden so that what the student solves for? This seems better, but please you tell me what you think.
  3. Is it helpful to bring in both bottles to class, the crushed one and the non-crushed one?

9 thoughts on “Altitude Sickness

  1. I had to sleep on this one. Four thoughts.

    One, I’ve had the “Idiot! Nincompoop!” moment a million times and I’ll have it a million times more. It always sucks.

    Two, I think this needs video of the bottle crumpling over time. At the very least a well-staged photo of before and after side-by-side with (maybe) the sound of the thing crumpling up played behind as the crumpled bottle fades into view. The connection that “this is what happens when the bottle comes down the mountain” isn’t as visually or aurally explicit as it could be, in other words.

    Three, I’m not sure how happy to be about 44% error.

    Four, “What’s the elevation of Pike’s Peak?” is more compelling than “What’s the elevation of Dan’s bathtub?” I recognize the difficulty of doing this problem in reverse, though.

    Fun stuff, in any case. I’m not sure I could make this work in my class as anything but an awesome conversation piece, but it sure is fascinating.

    1. 1. Indeed!
      2. Video would be great. Or at least some sort of time lapse photo.
      Hawaii’s highest mountain is at 13,780 ft, and you could take a video of the water bottle at the top and go all the way down to sea level in a couple of hours. Colorado doesn’t provide this easy of a change (drive to death valley?). However I’ve never been to Hawaii and it doesn’t look like it’s in the cards soon.
      3. Isn’t this measuring the change in altitude from pikes peak to my home? I’d put the relative error at 71m/(4300m-91.4m) = 1.68%.
      4. Its not that difficult to set up the reverse. We just need someone to reshoot the whole thing. Oh well. I like the seed of the idea.

  2. Is it helpful to bring in both bottles to class, the crushed one and the non-crushed one? –> yes, definitely. would you give them the volume of the crushed bottle or ask them to find it or just give them the numbers? because before reading the much better archimedes method, i had thought of a different way to measure the volume involving massing the two bottles and then using some of the other gas laws, somehow taking into account the plastic by weighing the bottle filled with water (would this work?). i agree with Dan about putting intermediate steps on the way down. i love this though, i think it’s really cool!

  3. Are the elevation markings on the trail? Sometimes trails have that you could take a picture of the bottle beside each sign. Do a little displacement at each sign to show the change in volume size. I love this Dan, especially because the same thing happened to me while driving through the mountains the containers that my wife and I had in the car. I am definitely going to have to clean up a spot, and set up a camera while I drive to my parents house.

    1. This was a mountain that you could drive up. My phone has a GPS app though that gives the elevation. What starting and ending elevation do you have when you drive to your parents house? Some sort of time lapse setup could also be cool, take a picture every minute or something. Interesting.

  4. Ok well I’ll try and do it somehow. I am thinking before after shots, then film the drive down with a gps beside the bottle. I was thinking of throwing in another camera and filming a bottle full of water as well (to talk about why one bottle compresses and another doesn’t. Anything more I should do?

    1. Hmm. Bottle full of water would be nice too. Maybe a 2 liter soda bottle too? Although the stiffness of the plastic certainly affects how much it crushes. A balloon?

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