Question

what physical law/s were used during the "burns" of Apollo 13 to adjust its path/trajectory and get it home?

I know the thrusters used Newton's third law. What else?

Answer 1

action and reaction (force on exhaust equal and opposite to force on rocket.)

conservation of momentum
velocity of exhaust* mass of exhaust + velocity of vehicle * mass of vehicle after exhaust fired = original velocity of vehicle * original mass of vehicle

note velocity of exhaust negative as a rule

gravity G M Mearth/r^2 = Force on vehicle toward earth center

Answer 2

Note do not forget centripetal acceleration m v^2/r