So I'm developing this game with physics.
Every frame, the body accelerates at $+4\,\mathrm{m/s^2}$.
However, every frame, the body also sets its velocity to 90% of its original value, basically the game equalivalent of drag.
How would I determine the terminal velocity here, with math, and not guess and check?
So, your function is kinda odd, but idk if that's intended. Lets analyze it:
$$v(0)=v_0$$ $$v(1)= 0.9v_0 + 0.9*4$$ $$v(2)= 0.9(0.9v_0 + 0.9*4 + 4)$$ $$v(2)= 0.9^2v_0 + 4(0.9^2+0.9)$$ $$v(3)= 0.9*(0.9^2v_0 + 4(0.9^2+0.9)+4)$$ $$v(3)= 0.9^3v_0 + 4(0.9^3+0.9^2+0.9)$$
See the pattern?
$$v(t) = 0.9^tv_0 + 4\sum_{t=1}^{t}0.9^t$$
So far so good, but notice that the $0.9^tv_0$ will converge to zero as t increases no matter what and the sum $ 4\sum_{t=1}^{t}0.9^t$ will converge to $0.9/(1-0.9)=9$ as $x-> \infty $. So, the terminal velocity will be
$$v_{ter} = lim_{t->\infty} (0.9^tv_0 + 4\sum_{t=1}^{t}0.9^t)= 4*9 = 36m/s $$
So if you accel and the correction parameters are always these two, the terminal velocity will always be 36 m/s for everything. In a general sense, if you can change the parameters, the function will be:
$$v(t) = C^tv_0 + a\sum_{t=1}^{t}C^t$$ $$v_{ter} = a \frac{C}{1-C}$$
Where "a" is the acceleration and "C" is the correction factor and $0<C<1$
EDIT: Just as a clarification, one might point out that the equation for the velocity is dimensionally incorrect as $a$ has units $m/s^2$ and $C$ is dimensionless, but this is so because every point of the function is exactly multiplied by a $\Delta t$ of 1. For instance, for t=2
$$v(2)= 0.9(0.9v_0 + 0.9*4*(t-(t-1)) + 4*(t-(t-1))$$
The $(t-(t-1)$ will reduce to 1 at all times, but it has units of time, these units are incorporated in the acceleration of 4m/s²
