Projectile Motion

From ground: R = v₀²·sin(2θ)/g. Max at θ=45°. From height h₀: R = vx·(vy + √(vy²+2gh₀))/g.

No, this ideal model ignores air drag. For real applications (ballistics, sports) consider aerodynamic drag.

For a launch from the ground the range is the square of the initial speed times the sine of twice the angle, divided by the acceleration of gravity. It comes out largest at forty-five degrees.

No, the model is an ideal one and ignores air resistance. In real cases, from ballistics to sport, drag cuts the range below the theoretical value, the more so the higher the speed.

Yes. Give a starting height and the time of flight grows along with the range, because the projectile keeps falling past the level it set off from before it reaches the ground.