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A Integration of velocity and thrust angle equation

  1. Oct 23, 2016 #1
    This is from a physics textbook, a chapter on rocket launch velocities, but really the question is how to integrate the first equation to get to the next.
    The way I was approaching it was like this:
    From
    ## V \frac{d\gamma}{dt}=-g \cos \gamma##
    Integrating from ##t=0## to some ##t##:
    ##\int_{t_0}^t \frac{1}{\cos \gamma}\frac{d\gamma}{dt} dt=-\int_{t_0}^t \frac{g}{V} dt##
    ##\int_{t_0}^t \frac{1}{\cos \gamma}d\gamma =-\int_{t_0}^t \frac{g}{V} dt##
    Then, using a standard integral for ##\sec \gamma##:
    ##[\ln (\tan \gamma + \sec \gamma)]_{t_0}^t = -\int_{t_0}^t \frac{g}{V} dt##
    ##\ln (\tan t + \sec t) - \ln (\tan t_0 + \sec t_0) = -\int_{t_0}^t \frac{g}{V} dt##
    ##\ln \frac{(\tan t + \sec t)}{(\tan t_0 + \sec t_0)} = -\int_{t_0}^t \frac{g}{V} dt##

    I think this is correct so far, but I don't see how to get the result in the book. Not least because the book keeps the gamma term.
    Can anyone help?
     

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  3. Oct 23, 2016 #2

    BvU

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    Does the book tell you what ##\gamma## is ? Because you don't tell us !
    And posting eqn (7.2) would help too
     
  4. Oct 23, 2016 #3
    Equation 7.2 is:
    ##MV\frac{d\gamma}{dt}=F\sin (\alpha + \delta) - Mg \cos \gamma + L + \frac{MV^2}{r}\cos \gamma##
    Where g is the local gravitation, which varies with r, ##\gamma## is the angle between the local horizontal and the velocity vector, and L is the lift.
    Most of these terms are left out in the simplified version I posted above, and I didn't think I needed anything but maths to integrate the first equation to get to the second...but maybe I do need some information or physical understanding to get to the final equation.
    Thanks in advance for your help
     
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