Projectile motion of a moving target

[shibe]

Homework Statement

A ship is steaming at 30 km/h on a course parallel to a straight
shore at a distance of 17000 m. A gun emplaced on the shore
(at sea level) fires a shot with a muzzle speed of 700 m/s when
the ship is at the point of closest approach. If the shot is to hit
the ship, what must be the elevation angle of the gun? How
far ahead of the ship must the gun be aimed?

Relevant Equations

v=at

this is problem #57 from chapter 4 of "Physics for engineers and scientists vol1" ,it was solved in the book by a method the author calls "successive approximation, he first calculated the flight time of the projectile and then "corrected" for the ship's displacement. i know this is a perfectly good method, but i wanted to directly describe the motion of both the ship and the bullet (relative to the bullet's point of projection) and then solve for t at which their displacements were equal. this initial attempt resulted in:
ship:
$$x(t)= \vec i(17000)+\vec j(0) + \vec k(\frac{25t}{3})$$
bullet:
$$x(t)=\vec i(700\cos\theta)+\vec j(700\sin\theta)+\vec k(0)$$

where I,J,K were defined as the unit vectors in the direction of the ship from the gun, the direction upward, and the direction in which the ship travels respectively.

But this had to be incorrect, since the projected bullet actually travels in a 2D plane connecting the origin and its point of impact(with the ship) such that the resolution of the bullet's velocity in that plane does not yield vectors that were perpendicular to the velocity of the ship (which i previously termed k). This is the main obstacle to my much desired displacement(t) functions.

i have tried to make the plane in which the bullet travels and its relation to the ship's velocity vector clearer with a pathetically(sorry) drawn picture.
hopefully someone can help in the construction of displacement(t) functions with a decent explanation i can follow. or maybe deliver the bad news explaining why its unreasonable and i should just stick with the author's method. thanks

 

[haruspex]

Several things wrong with your shell (bullet?) equation.

• You have a position on the left but velocities on the right.
• You need to allow a k component of launch velocity.
• You forgot gravity.