How Does Quadratic Air Resistance Affect a Puck on an Incline?

[Habeebe]

Homework Statement

I kick a puck of mass m up an incline (angle of slope = θ) with intial speed v₀. There is no friction between the puck and the incline, but there is air resistance with magnitude f(v) = cv². Write down and solve Newton's second law for the puck's velocity as a function of t on the upward journey. How long does the upward journey last?

Homework Equations

F=ma=m*\frac{dv}{dt}

According to Wolfram Alpha (I use this later):
\int \frac{dx}{a+bx^2} = \frac{arctan(\frac{\sqrt{b}x}{\sqrt{a}})}{\sqrt{ab}}

The Attempt at a Solution

I set the axes so x is along the ramp in the direction v₀ and y is normal to the ramp upwards. This gives force and acceleration in the x direction only.

F=weight+resistance=-mg*sin(θ)-cv²
a=\frac{dv}{dt}=-gsin(θ)-cv²/m

Separation of variables gets me to:
\frac{-dv}{gsin(\theta)+\frac{cv^2}{m}} = dt

I didn't know offhand how to do the integral, and it looked fishy, so I Wolfram Alpha'd it to see if I get something that makes sense before I figure out the method. Using that solution I with limits of v from v₀ to v and t from 0 to t I get:

\frac{ arctan(v*\sqrt{\frac{c}{ mgsin\theta }}) } {sqrt{\frac{cgsin(\theta)}{m}}} |^{v}_{v_0} = -t

And this is a jumbly mess. I can't really tell if I'm right or not because I can't identify intuitively what parts of the expression on the left stand for what. My gut feeling is that this can't be right because the answer is so absurdly ugly.

I also tried using \frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=\frac{dv}{dx}v on my original equation, ultimately getting:

\frac{cv^2/m+gsin\theta}{cv_{0}^{2}/m+gsin\theta}=e^{-2cx/m}

But this is a function v(x(t)) and I'm not really sure how to go about solving that for v(t).


Thanks for the help.

 

[haruspex]

\frac{ arctan(v*\sqrt{\frac{c}{ mgsin\theta }}) } {sqrt{\frac{cgsin(\theta)}{m}}} |^{v}_{v_0} = -t

Looks fine to me. You are asked to obtain v as a function of t, so a few steps to go yet.
For the time to reach highest point, what will you put for v?

 

[Habeebe]

v=0. It looks really easy to solve for (atria), I just need to know that I'm on the right track. This is the first class that I've had where the answers come out this ugly.