As My Mind Keeps Me Awake

Before I went to bed - as I was standing outside - I heard the sound of a multi-engine jet flying overhead, which is a common occurrence. In the past, before I moved to under a flight path, I never really paid attention to jets. After I moved, I noticed how I far the jet was from the apparent location that the sound revealed. It made me think of why, so as I thought, it came to me that at their altitude, and with their speed, the sound was arriving long past the current position of the jet. I had to look some things up, and built a spread sheet. 

Using the typical speed of commercial airlines, and their typical cruising altitude, I could use trigonometry to determine the angular difference between the sound and the location. While the parameters were basically wild-ass guesses, observation revealed the angular distance between the sound, and the jet, was pretty close to what was calculated. Still, on some occasions during the day, I couldn't find the jet, although I could hear it. This leads to the reason for this post, which became a compelling task when I woke at 1:30 am and couldn't fall back asleep. 

The atmosphere was mostly clear right after sunset, but the jet I was trying to find was pulling a contrail. As followed the contrail to the jet, faint navigation lights showed its location. From what I could see, the distance between the sound, and the jet, was greater than what I usually observe. This led me to realize the jets I couldn't find in the day were flying at an altitude where binoculars would be required to observe them; if they sky was really clear, and I managed to find them with binoculars. So, I'm thinking they're military jets. They're allowed higher altitudes than commercial flights. I might be wrong, but judging from the sound (which is fainter), and the fact even military transports are limited by air speeds similar to commercial transports, I think I'm right. 

I know some might be thinking calculus is more appropriate for calculating such things; and I think they're right. I don't know calculus, although I could probably learn it in some rudimentary way...if I was so inclined, which brings me to the second reason for this post. 

I was thinking of how Isaac Newton plotted the motion of planets. I can understand the method, but the demand for accurate timekeeping is necessary, and I wondered how that was done during his lifetime. I looked it up. It was using water-clocks, which are considered very accurate. 

Now I probably need to try to sleep, but I don't feel like it. I'll make a cup of coffee and see if that helps. If not, I'll wait a few hours and cook breakfast.