1. Not finding help here? Sign up for a free 30min 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!

Separating vector differential equation into components

  1. Sep 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Write down the component form of the differential equations of motion of a projectile if the air resistance is proportional to the square of the speed. Are the equations seperated? Show that the x component of the velocity is given by

    [tex]\dot{x}=\dot{x}_0e^{^-\gamma s}[/tex]

    where s is the distance the projectile has traveled along the path of motion and [tex]\gamma = c_2 / m[/tex]

    2. Relevant equations



    3. The attempt at a solution

    So, the differential equation in vector form is

    [tex]m \frac {d^2r} {dt^2} = -c_2\vec{v}|v| -gk[/tex]
    [tex]\frac {d^2r} {dt^2} = -\gamma\sqrt{V_x^2+V_y^2+V_z^2}(V_xi+V_yj+V_zk) -gk[/tex]

    so x in particular is:

    [tex]\ddot{x}=-\gamma\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\dot{x}[/tex]

    But this isn't separable, making things very difficult. I do think that

    [tex]s = \int_a^b|r'(t)|dt = \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}[/tex]

    or something very close to that. Yet I'm still not sure how I'm supposed to proceed. Maybe I'm getting something fundamentally wrong in the setup? Something else that's obvious? I'm fairly new to differential equations..
     
  2. jcsd
  3. Sep 17, 2016 #2
    Along x-axis, we have: ##m\ddot{x}=-\mu \dot{x}^2##
    Then ##-\mu(\frac{dx}{dt})^2=m\frac{dv}{dt}##
    Then ##-\mu v.dx=m. dv##
    Solve this equation you will have this result
     
  4. Sep 18, 2016 #3
    Thanks, this helped a lot!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Separating vector differential equation into components
  1. Vectors components (Replies: 3)

  2. Vector Components (Replies: 2)

  3. Component of a Vector (Replies: 5)

Loading...