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The problem I have is how the equation for velocity was found. I understand that Fnet = mg - pVg - bv, which is basically

**net force = weight - buoyant force - viscous resistance,**but I don't get how the velocity (equation below) was derived in terms of time, terminal velocity (Vt), and initial velocity (V0).

Is there a mathematical proof for this? Or was the equation created based on trial and error? If you could, could you also please explain how the equation for distance (image below) was also obtained? Thanks!