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You wake up with a gentle breeze on your face. Opening your eyes, you suddenly find that you are no longer at your home. You look around and find that you are on the roof of a very high building with no idea how you got there. You find a door, but it is locked.

You notice a paper sticking out of the bottom of the door. You open it and read-

Hey! I hope you enjoyed your morning surprise. As you might have already noticed there is only one way out of this roof and it is locked. If you want to get out you have to find out the height of the roof from the ground. Write the answer on the piece of paper and pass it under the door.

However, I'm in a good mood. So, I'll accept the answer as long as it is within 10% of the accurate height. I've also left some tools for you to use.

Good luck.

- Your prankster friend, Jason

Knowing that Jason only pulls the roughest tricks, you know there is no easy way out of this. You go over to the railing. You can see the ground below, but you are bad at estimating distances.

You look around and find a pendulum and a stop watch. These are what Jason left for you. There is nothing else in the roof that you can possibly use. You pick up the pendulum with your hand. It's quite heavy and has about a 20cm long string. Hum... pendulum and stopwatch, maybe I can do it, you think.

How do you find the height?

To prevent "loophole" answers the following rules hold:

  1. No, you don't know the height of the building.
  2. There is no way of communication except that piece of paper.
  3. The door is not breakable
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  • $\begingroup$ How long is the string on the pendulum? $\endgroup$
    – Rand al'Thor
    Aug 17, 2015 at 22:18
  • $\begingroup$ @randal'thor question edited, about 20 cm. $\endgroup$
    – Rohcana
    Aug 17, 2015 at 22:23
  • $\begingroup$ And we can assume the building is higher than 20 cm? :-P $\endgroup$
    – Rand al'Thor
    Aug 17, 2015 at 22:26
  • $\begingroup$ Could you throw the pendulum at a passing stranger to get their attention and have them unlock the door from the other side? $\endgroup$
    – 2xedo
    Aug 17, 2015 at 22:27
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    $\begingroup$ This reminds me of the infamous Barometer Question. $\endgroup$
    – f''
    Aug 17, 2015 at 22:30

5 Answers 5

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Assuming the building isn't very tall, you could drop the pendulum off of the side of building and start the stopwatch. Listening for when the pendulum hits the ground you could measure the time it takes to fall. Then you can do math to find the height based on time, assuming Jason hasn't taken this prank too far and changed gravity. So I guess it being a pendulum is irrelevant?

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  • $\begingroup$ That fast!!! I thought the pendulums would distract people. $\endgroup$
    – Rohcana
    Aug 17, 2015 at 22:30
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    $\begingroup$ Use spoiler tags. $\endgroup$
    – Rohcana
    Aug 17, 2015 at 22:32
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    $\begingroup$ That assumes the building is short enough to neglect terminal velocity and speed of sound. Given the words "very high building" in the question, I reject this in favor of a ridiculous loophole answer. :P $\endgroup$
    – Roland
    Aug 17, 2015 at 22:34
  • $\begingroup$ You don't even need a pendulum, you can just drop a shoe. $\endgroup$
    – user9771
    Aug 21, 2015 at 18:26
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A different approach:

Stand at the railing on whichever side of the building you can see the shadow it casts on the ground. You should see your own shadow at the top. Estimate the proportion of the total building+prisoner shadow is caused by you. This can be done most accurately by holding the pendulum string at a distance from your body, parallel to the ground (creating similar triangles) so that the top and bottom of the string align with the top and bottom of the shadow. Then pinch the points where the top of the building is. Fold the string to determine the ratio. Use the shadow of the railings to work out how much of your legs is lost in the shadow of the edge of the roof. Use the known length of the pendulum string to measure this point on your leg. Subtract that from your own height (which you should know). Using the ratio you estimated earlier, scale up the remainder of your height to determine the height of the building. Use the heavy pendulum to smash the stopwatch to get a sharp piece of metal or plastic to pierce your skin so that you can use your blood to write the answer on the paper.

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You can

lean over, count the number of floors, then multiple it by the average height of each floor (roughly about 3 meters).

Admittedly, this method might not be accurate within 10%. It also rely on you being able to lean far enough to count the number of floors.

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  • $\begingroup$ Interesting answer. Although average height might be vastly different for different buildings. $\endgroup$
    – Rohcana
    Aug 18, 2015 at 1:00
  • $\begingroup$ It's usually within 10% of 3 metres, though. $\endgroup$
    – user88
    Aug 18, 2015 at 2:59
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Maybe you could

Measure the period of the pendulum with the stopwatch and use the formula for the approximate period of a pendulum at small angles to find the acceleration g due to gravity(put the length of the string over the quantity of the period divided by 2 pi quantity squared). You can find how much it differs from the 9.81 m/s^2 at the Earth's surface and calculate from there using Newton's gravitational force approximation formula.

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  • $\begingroup$ This was the answer which was implied (as a red herring) in the question, but is incorrect. The length of the string is not long enough to get an accurate measure. And even if you could get one, you only know the distance from the center of the earth. Without knowing the distance between the center and the surface you can't measure the height. The height of the building is negligible to the radius of the earth, So, the difference of g on the surface and on the roof is too small to get an good estimate of the height. Also, g may vary from 9.78 - 9.82 depending on your position on the earth. $\endgroup$
    – Rohcana
    Aug 21, 2015 at 15:27
  • $\begingroup$ Ah darn I had assumed by very high you meant not negligible which is why I thought the dropping the pendulum off wouldn't work, because it would be too quiet. Like I wouldn't expect to be able to drop something off the Burj Khalifa and hear when something hits the ground because it would dissipate too much, especially since it gets windier the higher up in the atmosphere you go. And I didn't think seeing the ground meant anything because you can see the ground from however high up if it's clear. You should add something about still being below the clouds :) $\endgroup$
    – maxx12345678
    Aug 21, 2015 at 17:52
  • $\begingroup$ It did say that "you can see the ground below" ;) $\endgroup$
    – Rohcana
    Aug 21, 2015 at 21:41
  • $\begingroup$ Well like I said in my comment above, you can see the ground from however high up. You can see the ground from space if it's clear :) $\endgroup$
    – maxx12345678
    Aug 25, 2015 at 13:36
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From Wikipedia:

Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is...

Therefore,

we can interpret "height of the roof from the ground" as "vertical distance from the lowest point on the roof to the highest point on the roof, as observed from the ground". I hope your relative velocity with the ground is non-relativistic, so the height should be the same when you observe it. If the roof is flat, simply pace off a distance from the highest point, or measure multiples of your own height, then use trigonometry to determine the height of that highest point.

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