The Problem: A particle moves in a medium influenced by a retarding force mk(v^3+va^2), where k and a are constants. Show that for any initial velocity the particle will never move more than (pi)/(2ka) and that it comes to rest only for t -> infinity Attempt to solution: I know I have to integrate the retarding force [F=-mk(v^3+va^2))] so that I can solve for v. Later, I have to solve for the constant C of the indefinite integral. Afterwards, I have to integrate to find the position. To show the first part, I believe I have to find the value of the position as t-> infinity. To show the second part, I believe I have to find the value of the velocity as t->infinity. My problem: I cannot solve for v I have solved the integral of the retarding force using trig substitution (v=acot(theta)) (dv=-acsc^2(theta)d(theta)) My final answer is: t=(-1/ka^2)(ln(v/(sqrt(v^2+a^2))+C If I use the quadratic equation to solve for v, I get imaginary numbers.... Am I making a mistake in my calculus or algebra? or Am I missing a physical concept?