THE FOLLOWING APPROACH FAILED. IN THE NEXT UPDATE I WILL EXPLAIN WHY.

I struggled quite a time to find out how to calculate a movement of a particle in a field. I want to have as parameter step size (s), initial speed (v) and the acceleration (a) by the field. My approach is to move the particle through sections . The particle has an initial speed of v (in red). With this speed v I calculate an equivalent acceleration av to move the particle through the section with out any acceleration. A particle will cross a section even if there is no field to accelerated it when the particle has at the start of the section already a speed. Then I add the equivalent acceleration av with the accrual acceleration a by the field. The result is total acceleration atot. To cross a section the time ts is necessary with the known length s and the total acceleration atot. At the end of a section I collect the total time and the total distance of the particle. For the next section the procedure is the same with on exception the speed of this section is the speed of the section before.

The diagram on the right shows the values of the table left. You can see that the particle has a speed of 1m/s a the beginning (step 0). At the next steps 1,2,3 the particle will accelerated with 3 m/s^2. The speed v goes up. In the step 5,6 the speed of the particle will decrease by the acceleration of -5m/s^2 to the speed of zero.