But let's check the feasibility of this with a quick back-of-the-envelope calculation. For any realistic earthbound circumstance, we can assume that the pressure drops linearly with altitude; see the chart below. For reference, Denver, Colorado is at about 1600m, and the highest town in Great Britain is under 500m; we won't find much in the way of tall buildings above an altitude of 4000m.
("Atmospheric Pressure vs. Altitude" by Geek.not.nerd - Own work. Licensed under CC0 via Wikimedia Commons)
So what would our barometer tell us? Since we're just investigating the feasibility here, we're going to round things a little to make the math easy. Don't worry, no truths were harmed during the making of this calculation:
- For every 1000m of altitude gained, the pressure drops ~10kPa, the chart tells us (at least, over the range we are concerned with).
- Thus for each meter, the pressure drops ~10Pa; that's 0.1hPa. (1 hectopascal or 1hPa = 100Pa, and is the modern unit equivalent to millibars, commonly used in meteorology.)
Fortunately there are two other possible ways to use a barometer to determine a building's height:
- Drop the barometer off the top of the building and time how long it takes to hit the ground below. For a building in the range of 100m to 200m, timing accurate to 0.01s would give a precision of around 0.5m, so we will probably want to use some kind of electronic timing device rather than a hand-operated stopwatch. (With manual timing we could reasonably only count on a timing accuracy of 1/10s, which by coincidence converts to about the same height accuracy as the digital barometer.) For example, we could have synchronized clocks at top and bottom, an electromagnet release that records the start time and a sound-activated circuit to record the stop time. For a heavy barometer, we can ignore air resistance.
- Find the building custodian and say to him "If you can tell me how tall this building is, I will give you this lovely barometer". This is definitely my favorite solution.