Hey guys, its me again! This time, I was reading about projectile motion in my textbook, and noticed that they didn't prove some of the claims presented about this topic. So I decided to try and prove it myself as a challenge. I included it here as a PDF. It's a pretty short and simple proof, so if you are feeling bored and want to help me out, please check it and see if you can find a problem! I also included a challenging question at the end of the proof to test you guys/provide some mild entertainment. (Again, probably only appropriate for a period of extreme boredom...)
I can do your initial proof plus your "challenging question" in about 4 or so lines using conservation of energy.
Quite frankly I'm impressed you managed to follow your own work in this... I've spent the last 30 minutes or so trying to follow your math, but I can't make much sense of it. All I can say is that I agree with boneh3ad that conservation of energy is the easy way to go. Just write out the equation of motion for the projectile, and you will see that it is quite symmetric - parabolic... hence the t^2 term. This happens frequently in mechanics problems when you neglect non-conservative forces.
For a gravitational potential of mgh, this potential doesn't change explicitly with time. Therefore the energy of the object is constant. (The explanation for this requires doing some lagrangian mechanics). Therefore, 1/2 m v^2 + mgh = constant So if the object has the same height, its velocity must have the same magnitude.