- #1
luisgml_2000
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Homework Statement
Within the gravitational field produced by a celestial body of mass M we want to send a projectile of mass m from (x1,y1) to (x2,y2). The M mass is placed at the origin of coordinates. If the flight time is T, what is the initial velocity vector that we have to give to the projectile? Is this vector unique?
Homework Equations
The usual equations of the Kepler problem, ie
[tex]l=mr^2\dot{\theta}[/tex]
[tex]E=\frac{1}{2}m\dot{r}^2+\frac{l^2}{2mr^2}-\frac{k}{r}[/tex]
[tex]r=\frac{\frac{l^2}{mk}}{1+\epsilon\cos\theta}[/tex]
Maybe the most relevant one is
[tex]dt=\frac{m}{l}r^2\,d\theta[/tex]
since this equation includes time explicitly.
The Attempt at a Solution
Using [tex]dt=\frac{m}{l}r^2\,d\theta[/tex] and the equation for the orbit I integrated from [tex]t=0[/tex] to [tex]t=T[/tex] but the angular integration, that although can be done analytically, it turns out to be quite difficult, so I think this is not the right way to go.
The special case of a circular orbit is solved easily but I think the problem has to be solved in general.
Thanks in advance!