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Solution to Exercise 8
A fully open, outstretched hand spans about 20 degrees of arc from tip of thumb to tip of pinky. (A closed hand spans about 10 degrees; a finger width is about 2 degrees. Handy rules to compute arc...) So, this jet plane has moved 20 degrees in 15 seconds, or (20/57.3) = 0.35 radians in 15 seconds. Since the plane is 35,000 feet away from me, we also know that if it travels one radian of arc, it's moved a linear distance of 35,000 feet. So its linear speed is v, where v = (35000 ft * 0.35 rad) / (15 sec) = 820 ft/sec. Since there are 5280 feet per mile and 3600 seconds per hour, v = (820 ft/s) * (3600 s/hr) / (5280 ft/mi) = 560 mi/hr. Does this make sense? A typical flight from New York to Los Angeles takes about 6 hours, and the distance is about 3000 miles, so the average speed is 500 miles per hour. During takeoff and landing the speed is much slower than that, so the cruising speed of the plane is probably between 500 and 600 mph. We're in the right neighborhood, and thus are confident that, with just a hand and a watch, we've figured out the speed of a jet plane at an altitude of nearly seven miles! |