Nonlinear Retarding force

In summary, the error was in integrating the ODE for the particle's motion. The correct equation would be \frac{dv}{dt}=-k(v^3+a^2v)-Ce^(2kta^2), where k is positive.
  • #1
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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 separate 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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  • #2
The error is clearly in the integration of the ODE.
 
  • #3
Presumably, though you didn't show it, you used partial fractions to write
[tex]\frac{dx}{k(v^3+ a^2v)}= \frac{A}{v}+ \frac{Bv+C}{v^2+ a^2}[/tex]

How did you integrate [tex]\frac{Bv+C}{v^2+ a^2}[/tex]?
 
  • #4
I used mathematica, though partial fractions is the obviously the way to do it by hand.
 
  • #5
dextercioby said:
The error is clearly in the integration of the ODE.

The first error is integrating the wrong ODE.

[tex]\frac{dv}{dt} = - k (v^3 + a^2 v)[/tex] would be better, since the form of the answer implies k is positive.
 
  • #6
AlephZero said:
The first error is integrating the wrong ODE.

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

What is a nonlinear retarding force and how does it affect an object's motion?

A nonlinear retarding force is a force that is not directly proportional to an object's velocity. It can cause an object to accelerate or decelerate in a non-uniform manner. This type of force can significantly affect an object's motion, making it difficult to predict its movement.

What are some examples of nonlinear retarding forces?

Some common examples of nonlinear retarding forces include friction, air resistance, and drag. These forces can vary depending on an object's velocity and may increase or decrease as the object's speed changes.

How does the magnitude of a nonlinear retarding force change with an object's velocity?

In most cases, the magnitude of a nonlinear retarding force increases as an object's velocity increases. This is due to the fact that as an object moves faster, it experiences more resistance from the surrounding environment.

Can a nonlinear retarding force ever be beneficial?

Yes, there are cases where a nonlinear retarding force can be beneficial. For example, air resistance can help slow down a skydiver or a parachute, allowing for a safe landing.

How do scientists study and predict the effects of nonlinear retarding forces?

Scientists use mathematical models and simulations to study and predict the effects of nonlinear retarding forces. These models take into account various factors such as the object's velocity, surface area, and the properties of the surrounding medium to accurately predict its motion.

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