For a small personal project I'm looking at travel time of objects in a video game called EVE-Online.
I need to calculate time it will take object to travel from stand-still, constantly accelerating, until it reaches $x$ meters.
In this game objects velocity while accelerating in straight line is defined with equation:
Where $a$ and $s$ are object specific and unchanging for duration of acceleration.
The function is constructed in a way that $v(t)$ will approach $s$ (maximum speed of object), but never reaching it.
What I did to solve I would call a brute force approach: I calculated $v(t)$ for each second of simulation and summed it up until reaching $x$ (not exactly, as You will overshoot but my system will work fine with precision around one second).
Because I have to calculate this value for many thousand of objects it is impractical to perform this simulation for each and every single one due to computing time needed (I want my system to be relatively fast) and I'm looking for directly solving for $t$ needed to sum of $v(t)$ equaling to $x$.
Is there a way to solve this other than just sum up $v(t)$ at each second or fraction of a second until reaching designated goal?