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Kinematics of a gliding object

  1. Mar 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Finding the velocity of a gliding object with an initial thrust-velocity of V after time t.
    Constants are:-
    g: gravity
    a: constant for angle of attack, lift coefficient, air density and wing-area

    2. Relevant equations

    [tex]L \propto a*V^2[/tex]
    acceleration = v (dv/dt)
    V is the airspeed of the object



    3. The attempt at a solution
    I know lift is a force and has an acceleration. So thinking of Lift in terms of velocity-dependant acceleration and only focusing on Vy

    acceleration = -g + a*v^2
    [tex]v (dv/dt) = -g + a*v^2 [/tex]
    [tex]dv/dt = -g/v + a*v [/tex]
    [tex]dv/((-g/v) + a*v) = dt[/tex]

    [tex]\int_{v_0}^v dv/(-g/v + a*v) = \int_{0}^t dt[/tex]

    I havent done integration in a long time.. also Im confused about the notation
    Heres the equation I get
    [tex]\Big|_{v_0}^v (g*ln((a*v^2)-g) + a*v^2)/(a*v^2) = t [/tex]
    After this.. Im unsure on how to proceed further
     
    Last edited: Mar 29, 2010
  2. jcsd
  3. Mar 29, 2010 #2
    Try writing your integral as
    [tex]\int_{v_0}^v \frac{v}{-g+av^2}dv[/tex]
    Then substitute in u=-g+av^2
     
  4. Mar 29, 2010 #3
    thanks for the reply.. after integrating that i get [tex]\Big|_{v_0}^v \frac{ln(a*v^2+g)} {(2*a)} = t [/tex] ..which looks a lot more workable, but is the integration correct?
    and how do i proceed from here for coming up with a v = v0(t) equation? (i.e. a "Vf= Vi+at" -type equation where given an initial velocity I can come up with a velocity after time t)
    Im not asking about the actual math involved.. im just unsure as to what this:- " [tex]\Big|_{v_0}^v[/tex] " notation means, since it will determine the initial equation i need to simplify
    I tried the following way but..

    1) [tex]\Big|_{v_0}^v \frac{ln(a*v^2+g)} {(2*a)} = t [/tex]

    2) [tex] (v-v0) ( \frac{ln(a*v^2+g)} {(2*a)}) = t [/tex]

    3) [tex] (v-v0) = \frac{t*2*a} {ln(a*v^2+g)} [/tex]

    4) [tex] v = v0 + \frac{t*2*a} {ln(a*v^2+g)} [/tex]

    I get stuck again since I have an unknown [tex]v^2[/tex] in the fraction on the right side.. should i try another method from step 2 onwards or is the step wrong to begin with?
     
    Last edited: Mar 29, 2010
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