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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?