In any case, during this half-hour, sometimes my mind wanders. I start thinking. How high do the big ones go? How far away are they? Stuff like that.
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Recently, I watched a display where the sound got to me consistently 2 seconds after the flash. That means it was about 2,200 feet away from me when it exploded. (I was fairly far away from this show.)
In that recent display I saw, the flash was just above my palm. My hands are large, and my spread finger span is 10 inches, so just past the palm represented about 7 inches above the bottom fingertip. My hand was about 28 inches from my eye as I held it out.
Can you figure out the height? If you have an idea, try it before
OK. Remember any Geometry? Similar triangles? Well, we have a situation with two similar right triangles in our problem. The small one is: a horizontal imaginary line from your eye to your bottom finger (28 inches long); a vertical imaginary line in your hand (the 7 inches); and a hypotenuse connecting the ends. The big one is: a horizontal line from your eye to the launch point; a vertical line (the height that we want to know); and a hypotenuse connecting the ends (about 2200 feet).
OK. I'm about to make some approximations. You always need to be careful about any assumptions you make about such things. My thinking: My triangles are pretty long and flat, where the horizontal leg is much longer than the vertical. This should allow me to estimate that the horizontal leg is only a little shorter than the hypotenuse. I decide that about 2,000 feet is reasonable. Now we're in business.
My little triangle had legs of 28 inches and 7 inches, in other words 4 to 1. This same proportion must also be true of the similar large triangle. Since it's long leg is 2,000 feet, its short leg must be 500 feet. So this is the approximate height of the firework explosion.
For this method, we need one of those nasty EQUATIONS that Physics has so many of. This one is the relationship of distance and acceleration. D = 1/2 * A * t2. A is the acceleration due to gravity, which is 32 ft/sec/sec. t is the time interval (5 seconds). So D = 0.5 * 32 * 5 * 5 or 400 feet. Just counting the time from when it left the ground until it got to the top of its path, was enough to figure out how high it went!
That's not the same as the more accurate value of Method One (due to the assumption mentioned above), but it's reasonably close, considering the crudeness of the data collection methods. In any case, we have confidently determined that the height was about 400-500 feet, which we determined pretty easily (and probably even without paper!)
Before you did this, you might have concluded that you were not capable of figuring this out, or if you could, you would need a lot of accurate equipment and involved data collection. NOT TRUE!!
Our Answer (yours might be different!)
You could pace off the length of the tower's shadow on the flat (HORIZONTAL) ground. Of course, you must then know the average length of your pace. (When walking fast, mine is very close to three feet; when walking slower, it's about two and a half.) Now you have the shadow length in feet.
Stand straight up and see where the end of YOUR shadow is. Measure it (approximately) with your paces or hand lengths or lengths of a sheet of paper or shoe lengths or with anything else that you can measure the length of.
You now have a small similar right triangle, and you know both its leg lengths (your height and the length of your shadow). Since you know the horizontal leg of the tower triangle (its shadow), you can easily get the other leg (the tower's height)!!!
This is the case primarily because no one (and no textbook) showed them the incredible usefulness a moderate knowledge of Physics can be.
This series of lessons is meant to correct that situation. Students in High School or College Physics should be able to benefit from and EVEN ENJOY (!!) these Physics lessons. The lessons should help clarify the usages of a lot of those dry subjects and equations the teacher or professor tries to ram down your throat. These lessons are freely made available to teachers and professors for use as they desire, either on the InterNet or in the classroom.
(The preceding paragraphs appears in each lesson, in the event that someone happens to find a single lesson from this series as a result of a search-engine search.)
The High School Physics Lessons - Practical A of this series is:
C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago