# Calculating Orbital Velocity in 2D

Torstein Bjelland
Hi!

Imagine a planet is alone in a two dimensional universe. It has a mass mp. Suddenly, the planet's moon appears out of nowhere. It appears with a distance r from the planet, has a mass mm and velocities vx in the x-direction and vy in the y-direction. The distances rx and ry are also known. A gravitational force F from the planet acts on the moon. Fx and Fy 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 vx and vy 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!

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## Answers and Replies

Science Advisor
Homework Helper
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.

Torstein Bjelland