What is the Solution to the Retarding Force Problem?

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The discussion centers on solving the retarding force problem for a particle influenced by a force proportional to mk(v^3 + va^2). It is established that the particle will not exceed a maximum position of (pi)/(2ka) and will come to rest as time approaches infinity. Participants discuss the integration of the retarding force to derive velocity and position equations, with one individual encountering issues with imaginary numbers when applying the quadratic formula. The importance of verifying calculations and understanding the physical implications of the retarding force is emphasized. The conversation concludes with the realization of a previous mistake in the calculations, leading to a clearer understanding of the problem.
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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?
 
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Draw a free body diagram for the particle in motion.
Apply ##F=m\dot v##.
 
After the free body diagram this is what I obtained.
Since the only force acting on it is the retarding force, it equals: F= -mk(v^3+va^2) = m(dv/dt)
This is a velocity dependent force. Therefore, I integrated the force as dt = -dv/(k(v^3+va^2)
then, I used trig substitution for v = acot(x) and dv=-csc^2(x)
And after integrating I obtain:

t= (-1/ka2)(ln(v/(sqrt(v^2 +a^2)) + C

Is this correct so far?
 
skeer said:
After the free body diagram this is what I obtained.
Since the only force acting on it is the retarding force, it equals: F= -mk(v^3+va^2) = m(dv/dt)
This is a velocity dependent force. Therefore, I integrated the force as dt = -dv/(k(v^3+va^2)
then, I used trig substitution for v = acot(x) and dv=-csc^2(x)
And after integrating I obtain:

t= (-1/ka2)(ln(v/(sqrt(v^2 +a^2)) + C

Is this correct so far?
Have you checked that by differentiating to get the ODE?
 
haruspex said:
Have you checked that by differentiating to get the ODE?
Yes, I checked. I realize also what was my problem. Thank you all
 
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