Survey calculator

The Surveyor's Calculator contains useful formulas for  height(h) by drop time calculating ranges (a) and elevations (Y) based on measurable observations (`theta_1, theta_2,` and x).  It provides a height estimate (h) base on time of a vertical drop, and it also provides a simple Interpolation (linear) calculation.

If a projectile is launched at a speed uufrom a height H above the horizontal axis, gg is the acceleration due to gravity, and air resistance is ignored, its trajectory is

y=H+xtanθ−x2g2u2(1+tan2θ),y=H+xtan⁡θ−x2g2u2(1+tan2⁡θ),

and its maximum range is

Rmax=ugu2+2gH−−−−−−−−√.Rmax=ugu2+2gH.

I would like to derive the above Rmax,Rmax, and here's what I've done:

substitute (x,y)=(R,0)(x,y)=(R,0) into the trajectory equation;

differentiate the result with respect to θ;θ;

substitute (R,dRdθ)=(Rmax,0).

Recommendations

Additional formulas could be easily  Linear Interpolation  added to this calculator.  Please feel free to make suggestions by adding comments below.

projectile motion

motion of an object subject only to the acceleration of gravity

range

maximum horizontal distance a projectile travels

time of flight

elapsed time a projectile is in the air

trajectory

path of a projectile through the air

REFERENCE

[1]Projectile motion
Source: Wikipedia
URL: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/4-3-projectile-motion/