Calculation of the amount of energy required to slow down a falling body?

Question

Physics

In one of the tasks on codingame, it was necessary to land a satellite on the surface of Mars at a speed no higher than the specified one. The problem could be solved as a sharp braking immediately when the maximum speed is exceeded (it works). But it seemed to me that it was possible to let the falling satellite pick up speed, and start reducing the speed at the moment when the difference in kinetic energies would be equal to the work done.
So, there is a falling body with free fall acceleration gm - 3.711 (m/s^2), maximum speed - Vmax (40 m/s) current speed is given at each step - Vcurrent. The maximum acceleration from the engine is 4 m / s ^ 2, i.e. the difference between them - a = 4 - gm
According to my calculations, it was necessary to do

E1 - E2 = -A
(Vmax ^ 2 - Vcurren ^2) * m * 0.5 = - m * a * S
S = - 0.5 * (Vmax ^ 2 - Vcurrent ^ 2) / a

As a result, I should have obtained the height at which to start braking. Only this model did not work in the simulator, somehow the coefficient 0.5 did not fit. Now I sit and think where I made a mistake in deriving the formula.

Answer 1

Gregory, in your version the problem is divided into two parts. The work of the gravity force goes to increase the speed of the satellite from the maximum Vmax to the current Vcurren (assume that the accelerators are turned off). Change this energy you want to take fixed - ma S . Then, you need to offset this change in speed. At the same time, the satellite's engines will already be doing work against gravity. And this time the work will be the same.
So the satellite has traveled a distance of 2 S since the overspeed and your 1 / 2 factor disappears