- We are shooting a cannon from a moving platform (e.g., a ship). Platform velocity is $p$.
- We know where our target is in relation to us (vector $t$)
- We know the velocity of our shell ($|s|$)
- We are in possession of ballistics tables that for a given gun angle and shell velocity specify how far the shell is supposed to reach (and we are very quick in searching them so we have a function $f$ that returns angle $\alpha$ such as to hit a certain distance, assuming certain shell speed)
- The shell is relatively slow-moving so the platform velocity significantly affects aim, we cannot ignore it
- We assume the target stays still
- We want to find out where to aim our cannon so that we hit the target
- $s$ - initial vector for shell
- $p$ - platform vector for the platform that the shell is launched from ($p_y=0$)
- $t$ - vector to target ($t_y=0$, as we assume both are on the ground)
- $r$ - resulting vector for shell ($r = p+s$)
All vectors are 3D unless otherwise stated.
We are given:
- $|s|$ - initial shell velocity
- $p$ - platform vector
- $t$ - vector to target
- Function $f(|r|, d) = \alpha$ which for a given initial vector magnitude $|r|$ and distance $d$ we want to hit returns the angle $\alpha$ that hits the target
What we want to find:
- $s$ - initial shell vector
- Projection of $r$ onto the $xz$ plane is colinear with t (the resulting vector has to go in the direction of the target for us to hit it)
P.S. Seems simple enough, and I have been trying to solve this for a while, mainly by trying to reduce it to a 2D task by projecting it onto the plane where $r$ and $t$ vectors both lie but my vector math/geometry is too rusty.