We've all been there. It's 10am. Or 2pm. Or maybe even like 6:30pm. And you have some free time, and there's something you want to do, but your brain is so tired that you can't concentrate and you end up checking your phone every five minutes and the fun thing you want to do doesn't even feel fun, it's just this chore that you're dragging your brain to do becuase this is your *free time* and you're supposed to be *enjoying your hobbies*!
Let me stop you right there.
Go take a nap.
"But I'm not tired-" yes you are. "It'll mess up my sleep schedule-" we both know you weren't going to sleep on time anyway. "But it's a waste of time-" Forcing yourself to stay up wastes more time and I can prove it to you with math.
Here is a graph. X axis is time, Y is energy level, and the black line down the middle is the "Productivity threshold": the amount of energy you need to actually get stuff done. The blue represents a person who takes a nap. You can see the blue line drops to zero during the nap period. If you're sleeping, nothing's getting done. This is the "waste of time" you were talking about, right? The orange is the person who doesn't take a nap. Their enegy level just keeps going down. The yellow is the energy gain of Orange over Blue. The area is 56 X 46 = 2576. The blue area is the energy gain of Blue over Orange. The area is 178 X 88 = 15664. This is about 6x more energy, as the result of taking a nap. You'll notice that putting off the nap until later in the day will reduce the blue area, but won't increase the yellow area. So all you do by putting off the nap is reduce the benefit you can get after the nap. Now *that's* a waste of time!