Elliptical Orbits

Two astronauts are in the same circular orbit of radius R around the Earth, 180 deg apart.
Astronaut A has two cheese sandwiches, while Astronaut B has none. How can A throw a
cheese sandwich to B? In terms of the astronaut’s period of rotation about the Earth, how long
does it take the sandwich to arrive at B? What is the semimajor axis of the sandwich’s orbit?
(There are many solutions to this problem, assuming that A can throw the sandwich with
arbitrary speeds.)

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So I was given this question and at first, I was like, oh this seems easy only then to slap myself hard. My problem with this question comes in 2 different areas.

1. If extra velocity is given to an object in Orbit by the orbiting body, won’t it change the itself to a higher orbit, and therefore, it would never reach the second astronuat, yes I know that this orbit change is elliptical while experiencing velocity, but I am not sure anymore.
2. If A gives some velocity, momentum must be conserved so A would likely lose orbit.

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I am not sure how to approach it anymore. I know that the speed, relative to the astronaut
$$
0 < v < sqrt{frac{GM}{R}}(sqrt{2} – 1)
$$
But after that point, I am stuck. All I want is a point in the right direction. I feel really dumb because I feel that it is obvious, but I am missing a point. I am not sure how to apply it to retrieve the Semi-major axis, and Time taken. Any push in the right direction is appreciated.