Let us throw a particle of mass m from the north pole with a velocity v_0 forming an angle α with the surface of the earth. Considering only the effect of gravity and neglecting air friction...
a) What is the initial angular momentum l_0 when the particle is launched?
b) Compute the value of v_0 so that the particle will land precisely on the south pole. Express your result in terms of α, G (Gravitational constant), M the mass of the earth, and R its radius.
c) What is the eccentricity of the trajectory in terms of α only?
Hint: Use the equation of the trajectory r(θ) = (l_0^2/mk)/(1 - εcos(θ - θ_0)), and evaluate the initial angular momentum l_0. What is r(θ = ±π/2)? To compute the eccentricity, use the formula ε = √1 + (2El_0^2/mk^2) from the lecture notes.
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