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I was trying to create a better answer for this Stack Overflow question. I wanted to give the person a example of code using air resistance however every example on the net I find shows the formula : $$v(t) = \frac{1}{\alpha}\tanh(\alpha g t)$$ That assumes 0 starting velocity, I noticed they had "Parachute" variables so I assumed at some point in the future a parachute would be opened.

The problem I encounter is that I no longer have a starting velocity and time of 0, I tried to follow the derivation on the Terminal Velocity Wikipedia page but it has been too long and I do not know my calculus well enough anymore to change

$$t-0={1 \over g} \left[{\ln \frac{1+\alpha v^\prime}{1-\alpha v^\prime} \over 2\alpha}+C \right]_{v^\prime=0}^{v^\prime=v_t}$$

into

$$t-t_i={1 \over g} \left[{\ln \frac{1+\alpha v^\prime}{1-\alpha v^\prime} \over 2\alpha}+C \right]_{v^\prime={v_i}}^{v^\prime=v_t}$$ The farthest I got trying to find $v(t)$ was $$v(t) = \frac{1}{\alpha}\tanh(\alpha g t) , t < t_p$$ $$t - t_p=\frac1{\alpha g}({\mathrm{arctanh}(\alpha v)}-\mathrm{arctanh}(\alpha v_p)), t >= t_p$$

Can anyone help me out with the last steps, and please show the intermediate steps so I can learn how to do similar things in the future.

Also any help on finding $x(t)$ would be appreaceated too as I know I will likely have trouble finding that too.

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