Derive equation of trajectory of a body around a fixed body attracted by gravity

[gupta.shantan]

Homework Statement

There is a fixed spherical body of mass M whose center is to be taken as origin. Another body of mass m whose initial position vector $\vec{r}$ is given. This body is projected with initial velocity $\vec{v}$. Find the equation of trajectory of body with mass m around the body with mass M.

Homework Equations

Will the trajectory be an ellipse, just like the orbit of Earth around the sun?

The Attempt at a Solution

I tried solving the position using Newton's Law of Gravity. I also tried using the formula a = v dv/dx and integrating but was unable to reach a solution.

Any help is greatly appreciated...

 

[mfb]

Your problem is known as Kepler problem.

It is possible to derive the trajectory with Newton's Law of gravity, but this is an ugly calculation, at least ~2 pages long, involving polar coordinates, some substitutions and messy integrals.

Depending on the velocity, the radius and the masses M and m, the trajectory can be:
- an ellipse
- a parabola
- a hyperpola
which are all conic sections

 

[gupta.shantan]

thank you mfb
but i am willing to go through the messy mathematics. So can u please help me by giving me a link to where this Kepler problem has been solved.