Let's say we have a function:
$$v(t_n) = a\cdot(t-t_{n-1}) + v(t_{n-1}) \cdot d^{t-t_{n-1}}$$
Where
- $v$ = veolcity
- $t$ = time
- $a$ = acceleration
- $d$ = friction or damping
What is basically does is describe the velocity of an object which is continously accelerating but being breaked by a friction force proportional to its velocity. The curve looks like this:
As you can see from the $v(t_{n-1})$, it is using its previous result, so it is iterative if im not mistaken.
What I need is this function without iteration.
I've tried using a integral for this, but the results are not the same.
$$v_i(t) = a \cdot \left(\frac{d^t}{\log(d)} - \frac{1}{\log(d)}\right)$$
$$v(100) \approx 9$$
$$v_i(100) \approx 9.5$$
Oh, now wait a minute. When I decrease the time-steps for the first function, its results converge with the one using the integral.
Welp, sometimes you gotta write it down to fully understand.