# Nonlinear Retarding force

## Homework Statement

A particle moves in a medium under the influence of a retarding force equal to $$mk(v^3 + a^2 v)$$, 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 $$t \rightarrow \infty$$

## Homework Equations

Legrangian seems overkill, so I used Newton's.

## The Attempt at a Solution

$$\frac{dv}{dt} = k (v^3 + a^2 v)$$

then seperate the ODE and integrate

$$\frac{lnv}{a^2} - \frac{ln(a^2 + v^2)}{2a^2} = kt + C$$

multiply by 2a^2

$$2lnv - ln(a^2 + v^2) = 2kta^2 + C$$

use log properties and combine the natural logs

$$ln ( \frac{v^2}{a^2 + v^2} ) = 2kta^2 + C$$

exponentiate and carry constant down

$$\frac{v^2}{a^2 + v^2} = Ce^{2kta^2}$$

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

$$\frac{-a^2}{v^2+a^2} = Ce^{2kta^2}$$

use algebra to isolate v

$$v^2 = a^2 (1 - \frac{1}{1-Ce^{2kta^2}})$$

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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## Answers and Replies

dextercioby
Science Advisor
Homework Helper
The error is clearly in the integration of the ODE.

HallsofIvy
Science Advisor
Homework Helper
Presumably, though you didn't show it, you used partial fractions to write
$$\frac{dx}{k(v^3+ a^2v)}= \frac{A}{v}+ \frac{Bv+C}{v^2+ a^2}$$

How did you integrate $$\frac{Bv+C}{v^2+ a^2}$$?

I used mathematica, though partial fractions is the obviously the way to do it by hand.

AlephZero
Science Advisor
Homework Helper
The error is clearly in the integration of the ODE.

The first error is integrating the wrong ODE.

$$\frac{dv}{dt} = - k (v^3 + a^2 v)$$ would be better, since the form of the answer implies k is positive.

The first error is integrating the wrong ODE.

$$\frac{dv}{dt} = - k (v^3 + a^2 v)$$ would be better, since the form of the answer implies k is positive.

Yep, that was actually the problem. I went through it again and figured out what I did wrong. You end up with v = +/- a/sqrt(-1-Ce^(2kta^2).