# Problem 3 – Hint 5

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The will be no more hints for the Problem 3. The problem will be kept open for submissions until 20th March. Intermediate conclusions will be drawn each Sunday, 13:00 GMT.

Hint 4 already gave a full path to the solution. Let us just elaborate on those steps of that solution which might be more difficult to understand. We use the same enumeration as for the Hint 4. For convenience, we’ll be using system of units where $c=1$; we can later easily return to SI units by adding factors $c^n$ so as to have a dimensionally correct expression.
(2) From the energy conservation law, written in the lab frame, we can find the sum of the total mass of the photons $p$ sent in the first direction and the total mass of the photons $q$ sent in the second direction as the initial mass of the rocket minus the final relativistic mass of the rocket; this can be immediately calculated as a numerical factor times the initial mass of the rocket, $p+q=km$.
(3) The momentum conservation law in the lab frame can be used to express the modulus of the vectorial sum of the momenta of the photons, $|\vec p+\vec q|=rm$, where $r$ is a numerical factor.
(4) The relativistic invariants involving the 4-momenta pairs (a) and (c), as well as (b) and (c) will contain dot products of the momenta $\vec p$ and $\vec q$, as well as the dot product $\vec P\cdot\vec Q$ of the corresponding momenta in the rocket’s frame (at the moment when the direction was changed. Hence, the dot product $\vec p\cdot\vec q$ can be eliminated from the set of equations. The dot product of $\vec P$ and $\vec Q$ can be expressed in terms of the angle $\alpha$.