Derive the equation used by gunners that tells them at which angle to set their gun.?

Question

This is based on knowing the muzzle velocity of the gun, and the distance of the target. Hint: You will need a trig. identity.

Answer 1

You use the equations of motion
let v= muzzle velocity and th be the angle and d be the distance to target horizontally. I will assume the target is the same elevation as the gun muzzle

d=v*cos(th)*t

and the time to apogee of the shell is
0=v*sin(th)-g*t
or
t=v*sin(th)/g

the time to target is twice the time to apogee
so
d=v^2*cos(th)*sin(th)/g

2*cos*sin=d*g/v^2

2*sin(th)*cos(th)=sin(2*th)
sin(2*th)=d*g/v^2

j

Answer 2

2sin(θ)cos(θ) = sin(2θ)