Please submit the solution to this problem via e-mail to firstname.lastname@example.org. The next hint for the Problem 3 will be published at 13:00 GMT, 27th February 2022, together with the next updates of the intermediate results.
One possible way to solve this problem:
(1) Deteremine the final relativistic mass of the rocket from the given data.
(2) Write down the energy conservation law: the initial mass of the rocket equals to the sum of the total mass of the photons sent in the first direction, the total mass of the photons sent in the second direction, and the final relativistic mass of the rocket.
(3) Write down the momentum conservation law in a similar way.
(4) Write down the relativistic invariants (cf. Hint 1) for the following 4-momenta:
(a) the 4-momentum of the rocket at the moment when the direction of the engine was changed;
(b) the 4-momentum of all the photons sent in the initial direction (i.e. until the direction of the engine was changed);
(c) the 4-momentum of all the photons emitted after the direction was changed.
Note that the relativistic invariant involving only the 4-momentum (a) is trivial, and the relativistic invariant involving only the 4-momentum (b) is trivial. The remaining 4 combinations yield non-trivial and useful invariants, and two out of these four are expressed in terms of the angle \alpha.
(5) Using the conservation laws and invariants, express the angle \alpha in terms of a single parameter, for instance in terms of the mass of all the photons sent in the first direction as measured in the lab frame.
(6) Mininize \alpha.