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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).

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First, to be sure, in order for a rocket to pass a pole, you have to launch it at some angle to the pole.

For instance, if you are launching a rocket toward the North pole, it has to be aimed North-West, not directly North, which will affect its initial tangential component. Also, its velocity has to be high enough in comparison with the velocity of the Earth at the location of the launch - otherwise, it'll drift East too fast and will miss the pole.

At the moment the rocket is passing the North pole, it will be moving strictly South, i.e., its initial tangential velocity will disappear or, rather, it will be represented in the velocity pointing South. From that point on, the rocket will be drifting West and it new initial tangential velocity will be zero.

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