Galileo's Descendent
Galileo is purported to have tried to measure the speed of light by having two men climb two different hills, flash lantern lights back and forth, and measure the lapse in time, discounting for human reaction time. Of course, even if he had performed this experiment, it would not have worked, given that the speed of light is so fast that it can travel around the earth seven times in one second. However, Braggadocio, a great, great, great, great nephew of the honored Galileo, living in present-day New England, being somewhat arrogant, thinks he is smarter than his great, great, great, great uncle, and he can successfully perform the experiment because he has such quick reflexes.
Instead of using two nearby hills, Braggadocio decides to use the top of Mount Katahdin, 1 mile high (actually, it lacks 13 feet of being one-mile, but close enough for our purposes), at the northern terminus of the Appalachian Trail in Maine. He intends to flash the light from Mount Washington across the border in New Hampshire, 150 miles away, but realizes that numerous peaks of the Appalachian Mountains obscure the view in between, so he decides to set up somewhat eastward on Pleasant Mountain in Maine, not as high, at only a little over a third (.367) of a mile altitude, but also 150 miles away and without intervening mountains.
He waits for a perfectly clear night, uses powerful lamps that can be seen clearly over that distance, and sends his cousin, Pinocchio, whose reflexes are equally as fast as Braggadocio's, to the top of Mount Katahdin. The experiment fails, but not because the speed of light is too fast to measure. (Braggadocio still claims that he is fast enough to do it.) Why does the experiment fail?
(Hint: It is not because I did not use enough "great greats" in the preceding description!)
Answer:
Answer: The curvature of the Earth in between the two mountains blocks the path of the light.
Calculation: The amount of curvature of the earth between two points 150 miles apart can be calculated, using trigonometry, by imagining a triangle starting from the center of the earth and extending to the two points where the mountains are, and then drawing a line in between (call this line "A"). Drawing another line from the center of the Earth outward to divide this triangle in half, we end up with two right triangles. The hypotenuse of either triangle is the radius of the earth: 4000 miles. The short side of the triangle would be one half of 150 = 75 miles. The sine of the angle at the center of the earth in this triangle would then be 75/4000 = .01875. Using a calculator we find that the angle then is 1.07o. Multiplying the cosine of this angle by the hypotenuse distance of 4000 miles, gives us 3999.30 as the length of the long side of the triangle. Since the radius of the earth remains 4000 miles at that point, the "rise" due to the curvature of the Earth over the length of 150 miles is 0.70 miles. The height of our line of view from the straight line "A" at its midpoint would be the average of one-mile and .367 miles equals (1+.367)/2 = .68 miles. So even if the ground at the midpoint of this stretch were at sea level (which it is not) it would still obscure the view, extending .02 miles above the line of sight.