Suppose that a projectile which is subject to a linear resistive force is thrown

Question

Suppose that a projectile which is subject to a linear resistive force is thrown vertically down with a speed Vyo which is greater than the terminal speed Vterm. Describe and explain how the velocity varies with time and make a plot of Vy against t for the case that Vyo = Vterm

I have selected vertically down to be the positive y-direction. I found the sum of my forces to be gravity - air resistance (mg-cv). Im not 100% on this but from this I found acceleration to be g - (cv/m). From there I wrote acceleration as dv/ dt and set it equal to g - (cv/m). After cross multiplying I integrated -dv / (g-(cv/m)) and dt. After using a u substitution and letting c/m= a constant k, this gave me t= -(m (g-k)^2)/2c. After expanding, I got g^2 +k^2 v^2- 2gvk= -2tk. From here I don't know how to solve for v. Any help on how where I went wrong or how to solve the last equation for v in terms of t would be greatly appreciated.

Explanation / Answer

so mg - cv = m dv/dt

dv/dt = g - c v/m

dv/(g - c v/m) = dt

dv/( 1- c /mg v) = g dt

then integrate both sides

use that integral of 1/(1-ax) = - ln(1 - ax)/a

- ln( 1 - c v/mg)/( c/mg) = g t + C

- ln( 1- c v/mg) = ( c/mg ) ( g t + C)

ln( 1- cv/mg)= - c t/m + C

1 - c v/mg = A e^(-ct/m)

m g/c - v = A e^(-ct/m)

v = m g/c + A e^(-ct/m)

so as t goes to infinity, v goes to mg/c which is terminal velocity

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