Newton's second law states...
ΣF = ma
we can also write it in this form:
the forces working on the falling particle is the weight of the particle and the wind resistance which is variable to velocity, so we have (taking downward to be positive sense)...
where b is the coefficient to the wind resistance term; note: mg is constant since mass never changes and acceleration due to gravity never changes. So, lets use separation of variables method to solve this one..
the left side is easy enough to solve. The right side requires us to make the following substitution...
....
so we get...
solving the integral we get...
back substituting for u we get...
multiplying both sides by (-b) and exponentiating both sides of the equation we get...
this can be re-written...
e-bc can take on a new constant C2 and further simplifying the expression we get...
now we isolate for v...
the term (1/C2) can also take on another constant term C3 ...
further simplifying...
once again our term (C3 / b) can take on a new constant term K and our final answer is...
in this case: