Best launch angle and velocity: HANDS DOWN..

[Ether]

Originally Posted by Symph

I have revised the entire proof, sorry about the inconvenience, it should be sound now. However, considering the fact that I was thinking precisely that same thing last time I submitted this, I would encourage you to feel free to check my math again!
What say you, Ether?

This time around I didn't review all your steps again. Instead I did the following two things:

Thing1:

I took your "final answer" of V=29.53 f/s, angle=39 degrees, launch height = 3.61042 feet and calculated the trajectory from that. Your apex occurs at x=13.2555 ft and Rs is 11.3 ft.

Here's the equation for the trajectory using 29.53 f/s & 39 degrees:

y = -0.0305452*x^2 + 0.809784*x + 3.6104

(x is horizontal distance from launch point and y is height above floor)

Thing2:

I did my own calculation to find launch speed and angle to get the desired Rs=13.5 feet.

The result is

launch speed = 31.970 ft/sec at launch angle 35.55 degrees

The trajectory using those numbers is

y = -0.0237769*x^2 + 0.714568*x + 3.6104

max height above floor = 8.9792ft @ 15.0265ft from launch point

Bottom line: whatever you're doing is certainly close enough engineering-wise for FRC application. Mathematically, I suspect you are getting some round-off error due to the calculation method you used.

PS: You forgot to edit your original post. It still says "42.5 feet per second" and "13.5" ft scoring range.