Problem No 1
A ballistic missile is launched from the north pole, the target is at the latitude \Phi (>0, if northern hemisphere, <0 otherwise). At which angle (with respect to the horizon) the missile needs to be launched in order to have minimal launch velocity of the missile?
Remark: ballistic missile is like a stone: you give its initial velocity, and it moves due to inertia; you can neglect the air friction.
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Finally, what you may assume well-known and what not: things which are not in the formula sheet need to be motivated/derived. (You don't need to check – your typical high-school formulae are there.)