A rocket is trying to land on a planet. The mass of the rocket is $1\,\rm kg$, and the gravitational acceleration of the planet is $1\,\rm m/s^2$. The rocket starts the free fall at $20\,\rm m$ above the surface of the planet (initial velocity is $0$), and can use the thrust for $2\,\rm s$ (the force of thrust is $1\,\rm N$).
When is the most reasonable height at which the rocket uses its thrust for two seconds? (By the way, we ignore the loss of mass due to the use of the thrust.)
I solved the question, but I'm not satisfied. I don't quite understand it intuitively.
Someone said $W = Fs$, and since $F$ (thrust) is the same, when $s$ (distance moved) is the greatest, the work done by the thrust, to counteract gravity, would be the greatest. Therefore, the most reasonable height to start using the thrust is when the height at which the rocket would end using its thrust is when it reaches the ground (the calculation to find the actual value of the height is very complicated, so I'll skip (it's not the main point of my question).
When I first tried to solve this, I thought that the chemical energy of the thrust would be used to counteract gravity, and since the chemical energy of the the thrust does not change by the velocity at which it moves, I thought that the height at which the thrust is used does not matter, as the total energy (potential energy due to the gravity + the kinetic energy of the rocket - the chemical energy of the thrust) stays the same, the final velocity would be the same, but this is not the answer.
Can anyone please help me why I may be wrong?