constant velocity max velocity equation

Recall that when iterating movement we have the following where vi is the new velocity after the ith iteration, a is acceleration, f is friction, and vinit is the initial velocity:

v₁ = (vinit + a · dt) · f  
v₂ = (v₁ + a · dt) · f  
v₃ = (v₂ + a · dt) · f

Expanding, we get:

v₃ = (((vinit + a · dt) · f + a · dt) · f + a · dt) · f  
  = vinit · f³ + dt · a · (f³ + f² + f)

And in general, we can prove the general form:

vₙ = vinit · fⁿ + dt · a · ∑nk=1 fᵏ

Using the geometric series formula:

∑nk=1 fᵏ = (f(1 - fⁿ)) / (1 - f),   for f ≠ 1

Thus,

vₙ = vinit · fⁿ + dt · a · (f(1 - fⁿ)) / (1 - f)

So after n iterations, the velocity is given by the above. And when n → ∞, assuming 0 ≤ f < 1, then fⁿ → 0, and we get the steady-state velocity:

limn→∞ vₙ = dt · a · f / (1 - f)

Here is a link to see this in action on desmos