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.(adsbygoogle = window.adsbygoogle || []).push({});

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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# A Integration of velocity and thrust angle equation

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