From circular orbit to elliptical orbit

[physicsnmathstudent0]

Problem: a particle of mass m is in a circular orbit around a planet at a distance R from the center. The planet mass is M and it's radius is R_0.
What is the tangential impulse that will cause the particle to brush against the back of the planet? Describe the orbit.

The attempt at solution:

So I was reading Resnik's and Halliday's fundamental of physics to get a better understanding of this problem, and I found a similar problem in the sense that this ship was given a tangential impulse such that it's orbit changed from circular to elliptical, but in that problem the desired unknown was the new period in the new orbit. Anyway, I thought that that sample problem was useful to get a general idea of my problem.

So in the book it was illustrated that although the ship changes orbit, it always returns to the same point at which the impulse was given, so if I were to give my rocket an impulse such that it brushes (I don't now if this is the right word but English is not my language sorry) this planet's back, then that would mean that my rocket would change it's circular orbit for an elliptical orbit such that the perihelion would be when the rocket is at a distance R_0 from the center of the planet and the aphelion would be at a distance R (as I mentioned before I took this idea from sample problem 13.06 of Resnik's and Halliday's fundamental of physics fig. 13-17), so since it is an elliptical orbit it obeys R + R_0 = 2a, (1) where a is the semimajor axis, and we also know that the mechanical energy is E=-GmM/2a (2), but it also is the sum of kinetic energy and potential energy E = K + U, (3) where K= 1/2(mv^2), (3a) and U= -GmM/R, (3b), and since v would be the velocity just after it is impulse, then v = v_0 + delta v (4), then I substitute (4) in (3a), and I do all the algebra to find delta v. At least that's what I think, but I do have some questions regarding U and the fact that the rocket returns to the same point where it was when we gave it the impulse.

First, again, in the book it says that it takes the U just after it was given the impulse, but why is it valid to do that? And why does the rocket return to that point?

Thank you, I hope it's not too long

 

[Baluncore]

And why does the rocket return to that point?

Think of it as a mass or a satellite, not a rocket.
Before the impulse it was in repeating circular orbit.
After the impulse it is in a repeating elliptical orbit.
Both those orbits share a common point in space.

 

[physicsnmathstudent0]

Think of it as a mass or a satellite, not a rocket.
Before the impulse it was in repeating circular orbit.
After the impulse it is in a repeating elliptical orbit.
Both those orbits share a common point in space.

Thank you! Could you help me reviewing my logic in this problem, I'm not sure how to make clear that the delta v decreases the velocity, I think the sign is wrong

 

[Orodruin]

Thank you! Could you help me reviewing my logic in this problem, I'm not sure how to make clear that the delta v decreases the velocity, I think the sign is wrong

This should come out of your equations. If you have defined it as positive when increasing velocity you should get a negative result.

an elliptical orbit such that the perihelion

Pet peeve: Perihelion means the point closest to the sun … If you want to talk about the orbit around a general celestial object the word is periapsis.