# A Integration of velocity and thrust angle equation

1. Oct 23, 2016

### tomwilliam2

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?

#### Attached Files:

• ###### Integration.jpg
File size:
74.3 KB
Views:
42
2. Oct 23, 2016

### BvU

Does the book tell you what $\gamma$ is ? Because you don't tell us !
And posting eqn (7.2) would help too

3. Oct 23, 2016

### tomwilliam2

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.