(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant equations

Legrangian seems overkill, so I used Newton's.

3. 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Nonlinear Retarding force

**Physics Forums | Science Articles, Homework Help, Discussion**