Two astronauts are in the same circular orbit of radius \(R\) around the earth, \(180^{\circ}\) 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 \(\mathrm{B}\) ? What is the semi-major axis of the sandwich's orbit? (There are many solutions to this problem, assuming that A can throw the sandwich with arbitrary speeds.)