Homework Help: Horizontal Motion with Quadratic Drag

1. The problem statement, all variables and given/known data

Consider an object that is coasting horizontally (positive x direction) subject to a drag force f = -bv - cv^2 . Write down Newton's second law for this object and solve for v by separating variables. Sketc the behaviour of v as a function of t. Explain the time dependence for t large. (Which force term is dominant when t is large?)

2. Relevant equations



3. The attempt at a solution

I start with Newton's second law:

m(dv/dt) = -bv -cv^2

m(dv/dt) = -v(cv + b)

(-m/v)(dv/dt) -cv = b

dv/v + (cv/m)dt = (-b/m)dt

Then... do I integrate with respect to v for the dv terms and t for the dt terms? I can't figure out what this will give me with the second term on the right...

 

I get to the point where all my v terms are on one side so that I have:

(m dv)/[v(cv+b)] = -dt

From here when I try and integrate with respect to v on the left and t on the right, I'm doing a definite integral setting the limits between v and v0 and t and t0 so I get:

m[ln(cv^2 + bv) - ln(cv0^2 - bv0^2)] = -t

ln(cv^2 + bv) = -t/m + ln(cv0^2 - bv0^2)

Then if I take the exponential of both sides I still end up in a situation where I can't solve for v because there will be v on both sides... i.e

cv^2 + bv = e^(-t/m) + cv0^2 - bv0^2

How can I solve this for v?

 

How exactly do I integrate m(dv)/[v(cv+b)] ? The way I do it I consistently get natural log expressions which I am guessing is wrong because the function should tend towards 0 when I plot it...

 

Ah that clears things up. I see where I went wrong there, thanks for your help!