Half a dozen ways to measure the height of a building
Feel free to add to the list. It is a fun problem to try to think of more solutions for.
1. Boring and Simple: Hang rope from top.
2. Geometry: Use shadow of building vs. shadow of object with know height (similiar triangles).
If it is cloudy or night time you can instead eclipse the building with an object of known height standing known distance away. If there are no such object you could try…
3. Trig: Stand a known distance from base of building and find angle of inclination.
If you don’t have a way to measure angles you could try…
4. Calculus: Drop object from top of building, time it to hit ground, solve for height using equation which if not known can be found by integrating h”(t) = -32 twice with initial velocity zero and initial height the variable to be solved for.
If the height of the building is enough for air resistance to cause a significant error you could…
5. Embrace Friction: Drop an object with “imediate terminal velocity” such as a little action figure with parachute and time it to hit ground. You have to know what that velocity is of course. You could get it by dropping from height of known object. This approach is best done with repeated trials to smooth out the variation in the chute catching air pockets, etc.
6. Physics: Drop object from top of building, having it pass through a fastball sensor near bottom to measure speed. Use equality of initial potential energy with final kinetic to find initial height.
Suppose you are actually at Seven Falls National Park, Colorado trying to measure the height of the observation deck which is accessed by walking through a winding tunnel dug into a mountain and taking an elevator. Then you cannot drop an object from top or measure distance to base on ground level. Perhaps you could try…
7. Physics with Calculus: Get in the elevator at the bottom. Stand on a impractical to make but in theory possible hybrid between a scale and a polygraph machine. The needle of the scale has tape passing underneath it. Before the elevator is moving the scale registers your weight. As the elevator moves the reading will increase by exactly the amount of force the elevator is applying to your body. The needle records this on the tape. You can find acceleration by dividing the force of the elevator by your mass which can be found from scale. Thus the tape gives you an “acceleration vs. time graph”. Using calculus you can integrate this curve and use initial velocity = 0 to get a velocity versus time graph. Then integrate again (intial height = 0) to get a height versus time graph, which includes final height.