# How does an initial velocity affect the tangential velocity when a rocket cross the poles?

I'm trying to improve a piece of code I've written to allow for shooting rockets across the poles while being affected by the coriolis force. The code is written using RK4 since we are also looking at the drag forces.

The trajectory reference frame is located with origin $$(x,y,z) = (0,0,0)$$ at any given position by longitudinal and latitudinal angles. I've managed to write a method for calculating the coreolis force for whenever the trajectory stays with the x-axis in the North to South direction, and the y-axis in the West to East direction.

I initially assume that the canon has a relative velocity of $$v_T$$ which comes from the fact that its stuck to the ground while the earth rotates.

Whenever I fire the canon in the North/South direction, i simply subtract the initial tangential velocity with the new tangential velocity, for the given latitude.

My question is, how would I add the contributing initial tangential velocity, $$v_T$$ that my object has when I pass the pole? I have already added that the contributing tangential velocity changes direction when it happens (since the trajectory reference frame turns upside down).