# Newtonian particle problem with air resistance

Hey guys, first time poster but I'm a physics major and probably gonna stick around for a while to help people or get some help myself :P

Anyway, on to the problem!

## Homework Statement

A particle is released from rest and falls under the influence of gravity. Find the relationship between v and falling distance y when the air resistance is equal to γv3

F=ma=-mg+γv3

## The Attempt at a Solution

I've been able to solve this same problem for γv and γv2 but I can't seem to find an easy way to find
∫dv/(-g+v3)=t+C without Mathematica giving me an incredibly complex answer.

I don't want hand-outs, I want to solve this myself, but I don't have any idea where to go with this. Any tips on what I should be looking at to attempt a solution?

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TSny
Homework Helper
Gold Member
A difference of cubes will factor. So you might want to define a constant b so that you can write your denominator as v^3 - b^3.

I'm unsure how that would work? Should I define b as being the cube root of mg so I end up having

m*dv/(v^3-b^3)? I'm not sure how I would integrate from there.

rude man
Homework Helper
Gold Member
That integral is available in standard tables (e.g. CRC). It's messy but it's a closed-form expression. Have you tried to apply it to your integral? BTW in your integral the coefficient for v3 should be γ/m, not that that's a show-stopper. Also, I would choose y > 0 for t > 0 (just a sign change).

The problem doesn't ask for v(t), it just asks for the relationship between v and t. So I would argue that t(v) is an acceptable answer. Try it on for size anyway! :-)

gabbagabbahey
Homework Helper
Gold Member
I'm unsure how that would work? Should I define b as being the cube root of mg so I end up having

m*dv/(v^3-b^3)? I'm not sure how I would integrate from there.
Use partial fraction decomposition. You can decompose a fraction of the form $\frac{1}{(x-a)(x-b)(x-c)}$ as a sum of linear fractions $\frac{A}{x-a}+ \frac{B}{x-b} + \frac{C}{x-c}$ by a suitable choice of constants $A$, $B$, $C$.

Thanks guys, I got it with partial fraction decomposition.

I also realized I was heading the wrong way since I was finding the relationship between v and t while the question prompted to find the relationship between v and y. So I did a=dv/dt=dy/dt * dv/dy=v*dv/dy to solve the problem.

It wasn't too tough after that. Thanks for all the help guys, I really appreciate it