The Trans lunar injection burn transfers the craft from Earth low orbit to an earth orbit with an apogee that allows the craft to pass in front of the leading edge of the moon. At the point the engine is fired again to slow the craft again. This reduces velocity and drops it into lunar orbit.
It is this last brake manoeuvre I am interested in -
to drop into lunar orbit. The spacecraft has left Earth in direction, where the leading edge of the moon will be at the encounter a couple of days later. The Moon orbits Earth at a speed of abt 1 000 m/s. The speed of the space craft after leaving Earth and arriving Moon varies considerably from say 11 000 m/s to get away from Earth and say 600 m/s at the point where Earth's and Moon's gravity has equal effect and 2 400 m/s when arriving at the leading edge of the Moon and then increasing fast as the Moon pulls you towards its centre. To get into a stable orbit around the Moon you have to slow down to 1 500 m/s at exactly the right time and direction. If you brake too much you will continue to descend and must land at once. If you brake too little you will miss the Moon altogether and spin off into space and maybe return to Earth after an orbit around the Sun two years later. You also have to stay in the right plane - not veer off. NASA says you need 11 000 kg of fuel which you burn during six minutes to drop inte lunar orbit. It seems the brake force is constant so the deceleration will increase during the braking. Average deceleration may be 2.5 m/s² so you better be strapped in your seat. Everything loose inside the space craft will fly against the forward end of the space craft.
To win the Challenge you have to describe the space craft speed vector during the brake time, e.g. every second.
Reason why you do not land at once on the Moon Tin Tin Belgian 1953 style, i.e. you decrease the speed from 2 500 m/s to 0 at touch down is that this manoeuvre requires much too much fuel, which you cannot take with you from Earth, etc, etc. You get too heavy
According my calculations you will always run out of fuel regardless how you plan your manned space trip and as there is no way to fill up in space any manned trip is impossible. You cannot win my Challenge. This NASA 1969 Moon trip fantasy where you drop off parts of your space craft as you go along and only returns in a capsule with a heat shield dropping into the ocean is just a silly joke.