Relative to space, a spacecraft moving at velocity v in the y direction (y is upwards, x to the right) is getting closer to a planet moving at velocity u in the opposite y direction. The spacecraft does a hyperbolic trajectory with eccentricity sqrt(2) so that it ends up moving in the x direction.
When the spacecraft is far away from the planet, what is its relative speed to space?
Conservation of momentum
Conservation of energy
The Attempt at a Solution.[/B]
We got this problem in my analytical mechanics class.
Anyway, the Lagrangian for the two body problem is (taking M is the mass of the planet and m is the mass of the spacecraft):
There isn't any dependency of the generalized coordinate Xcm or time, therefore, according to noether's theorem, there is conservation of energy and momentum.
At this point, I assumed, that since the mass of the planet is much larger than the mass of the planet, mu=m. Also, the change in the magnitude of the velocity of the planet is negligible.
Assuming that the planet deflected in some angle theta (since the spacecraft couldn't have simply gathered momentum in the x direction), I got, from conservation of momentum in the x direction:
and in y:
but I don't know the masses or the angle theta..
From conservation of energy, with my assumptions, I simply get that the magnitude of v before and after is the same.
Not sure if my approach is right or it is supposed to be dependent of the masses or angle like I've written down or I missed something so I'd love some feedback and help with it.
Thanks and happy holidays.