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## Homework Statement

A particle moves in a medium under the influence of a retarding force equal to [tex]mk(v^3 + a^2 v)[/tex], where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater than pi/2ka and that the particle comes to rest only for [tex]t \rightarrow \infty[/tex]

## Homework Equations

Legrangian seems overkill, so I used Newton's.

## The Attempt at a Solution

[tex]\frac{dv}{dt} = k (v^3 + a^2 v)[/tex]

then seperate the ODE and integrate

[tex] \frac{lnv}{a^2} - \frac{ln(a^2 + v^2)}{2a^2} = kt + C[/tex]

multiply by 2a^2

[tex] 2lnv - ln(a^2 + v^2) = 2kta^2 + C[/tex]

use log properties and combine the natural logs

[tex]ln ( \frac{v^2}{a^2 + v^2} ) = 2kta^2 + C[/tex]

exponentiate and carry constant down

[tex]\frac{v^2}{a^2 + v^2} = Ce^{2kta^2}[/tex]

add and subtract a^2 in numerator to simplify, and then subtract the one

[tex] \frac{-a^2}{v^2+a^2} = Ce^{2kta^2}[/tex]

use algebra to isolate v

[tex] v^2 = a^2 (1 - \frac{1}{1-Ce^{2kta^2}})[/tex]

Now I am at the point where I can solve for the integration constant, but I don't see it giving me anything close to revealing a max distance of pi/2ka. Also, in my solution as t approaches infinity v = a, and not zero. Maybe I made an algebraic mistake?

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