Calculating Orbital Velocity in 2D

[Torstein Bjelland]

Hi!

Imagine a planet is alone in a two dimensional universe. It has a mass m_p. Suddenly, the planet's moon appears out of nowhere. It appears with a distance r from the planet, has a mass m_m and velocities v_x in the x-direction and v_y in the y-direction. The distances r_x and r_y are also known. A gravitational force F from the planet acts on the moon. F_x and F_y are known, as well as the angle α (see figure in attachment). All variables mentioned above are known.

Here is my question: How do you calculate v_x and v_y at any point in the moon's trajectory without knowing the semi-major axis, orbital period or what type of orbit the moon is going to enter in the first place?

Thank you in advance for all answers!

 

[Simon Bridge]

Hint: Newton's Laws of motion.
Rather than have the moon "suddenly appear out of nowhere" (which would probably destroy everything) just set up the conditions at t=0 and solve the differential equation.

Notice that you have to specify $\vec v(0)$ as well as $\vec r(0)$ ... if you want to insist that $\vec v$ must place the moon on an orbit, then it's direction must be tangent to an orbit and it's magnitude appropriate to get around the ellipse or whatever type of orbit you want.