Results 1 to 3 of 3

Math Help - Differential Equation direction fields

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    15

    Question Differential Equation direction fields

    The equation for a stone falling under gravity with air resistance,
    dv/dt = g - kv^2, v > 0,
    where k is a constant drag coefficient. Pay particular attention to the direction along the line v = sqrt{g/k} = vt. What is the significance of vt?
    Sketch solution graphs which start at t = 0,
    (i) for an initial velocity > vt and
    (ii) for an initial velocity < vt.
    I know about putting in values to draw the direction fields, but this question has thrown many people in our class and our lecturer said it was easy, so any help would be greatly appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,698
    Thanks
    454
    v_t = \sqrt{\frac{g}{k}} is terminal velocity ... note that a = \frac{dv}{dt} = 0 when v = v_t.

    for v < v_t , the speed will increase at a decreasing rate, with v \to v_t as t \to \infty.

    for v > v_t , the speed will decrease at a decreasing rate, with v \to v_t as t \to \infty.

    hope that helps with your sketch.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Quote Originally Posted by njr008 View Post
    I know about putting in values to draw the direction fields, but this question has thrown many people in our class and our lecturer said it was easy, so any help would be greatly appreciated.
    Hey, if I shoot a bullet straight up, how fast is it going when it hits the ground? Not too right? Same way if I start up there. If I shoot it down faster than terminal velocity, it will slow down, if I shoot it slower, it will speed up. So the plots look like the one I drew below: asymptotic to terminal velocity right? Here's the Mathematica code if you want to experiment with it:

    Code:
    line = Graphics[{Dashed, 
         Line[{{0, N[Sqrt[16/0.05]]}, 
           {5, N[Sqrt[16/0.05]]}}]}]; 
    slow = NDSolve[{Derivative[1][v][t] == 
          g - k*v[t]^2 /. {g -> 16, k -> 0.05}, 
        v[0] == 1}, v, {t, 0, 5}]
    slowplot = Plot[v[t] /. Flatten[slow], 
       {t, 0, 5}]
    fast = NDSolve[{Derivative[1][v][t] == 
          g - k*v[t]^2 /. {g -> 16, k -> 0.05}, 
        v[0] == 40}, v, {t, 0, 5}]
    fastplot = Plot[v[t] /. Flatten[fast], 
       {t, 0, 5}, PlotRange -> All]
    Show[{slowplot, fastplot, line}, 
      PlotRange -> {{0, 5}, {0, 50}}]
    Attached Thumbnails Attached Thumbnails Differential Equation direction fields-stone-fall.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Graphing Direction Fields for Differential Equations?
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: September 16th 2011, 05:33 AM
  2. tangent vector fields as linear partial differential operators
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 29th 2011, 11:35 PM
  3. Slope Fields/Differential Equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 29th 2011, 01:11 PM
  4. Replies: 2
    Last Post: August 15th 2010, 01:29 AM
  5. direction field-diiferental equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: January 17th 2009, 12:44 PM

Search Tags


/mathhelpforum @mathhelpforum