1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Quadratic air resistance on a ramp

  1. Aug 28, 2013 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations


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

    3. 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.


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

    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:

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

    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 [itex]\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=\frac{dv}{dx}v[/itex] on my original equation, ultimately getting:


    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.
    Last edited: Aug 28, 2013
  2. jcsd
  3. Aug 28, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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?
  4. Aug 29, 2013 #3
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted