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Main Question or Discussion Point
Hi!
I wonder how it is possible to accelerate a satellite using the gravity of a planet?
My amateur calculations would go like this:
Simplifying and considering a straight forward approach towards the planet where the end distance to core is r_{2} which of course will be grater than the planet radius which we may call R.
Using the basic fact that speed is the integral of acceleration we may (perhaps) write
[tex]\Delta v=a_r\frac{1}{m} \int_\infty^{r_2} Fdr[/tex]
and while
[tex]F=\frac{mMG}{r^2}[/tex]
this means
[tex]\Delta v=MG\int_\infty ^{r_2}\frac{1}{r^2}dr=\frac{MG}{r_2}[/tex]
Is this right?
I would however much more like a formula for the actual trajectory and I would very much like to try to derive it myself. But I need help.
First, what kind of coordinate system should I use?
I was thinking of cylindrical coordinates.
And there will be three angles. One for the gravitational force (gamma) and one for the speed direction of the vessel (alpha) and one for the release of the vessel (beta).
What do you think? Is it possible to derive the change of speed and beta using this?
I am attaching a drawing of this thought.
Take care!
Best regards, Roger
PS
I have tried to use energy conservation for this problem but has failed.
But when
[tex]Fc=\frac{mv_2^2}{r_2}[/tex]
equals
[tex]Fg=\frac{mMG}{r_2^2}[/tex]
There should be a release of vessel at
[tex]v_2>\sqrt{\frac{MG}{r_2}}[/tex]
Am I right?
I wonder how it is possible to accelerate a satellite using the gravity of a planet?
My amateur calculations would go like this:
Simplifying and considering a straight forward approach towards the planet where the end distance to core is r_{2} which of course will be grater than the planet radius which we may call R.
Using the basic fact that speed is the integral of acceleration we may (perhaps) write
[tex]\Delta v=a_r\frac{1}{m} \int_\infty^{r_2} Fdr[/tex]
and while
[tex]F=\frac{mMG}{r^2}[/tex]
this means
[tex]\Delta v=MG\int_\infty ^{r_2}\frac{1}{r^2}dr=\frac{MG}{r_2}[/tex]
Is this right?
I would however much more like a formula for the actual trajectory and I would very much like to try to derive it myself. But I need help.
First, what kind of coordinate system should I use?
I was thinking of cylindrical coordinates.
And there will be three angles. One for the gravitational force (gamma) and one for the speed direction of the vessel (alpha) and one for the release of the vessel (beta).
What do you think? Is it possible to derive the change of speed and beta using this?
I am attaching a drawing of this thought.
Take care!
Best regards, Roger
PS
I have tried to use energy conservation for this problem but has failed.
But when
[tex]Fc=\frac{mv_2^2}{r_2}[/tex]
equals
[tex]Fg=\frac{mMG}{r_2^2}[/tex]
There should be a release of vessel at
[tex]v_2>\sqrt{\frac{MG}{r_2}}[/tex]
Am I right?
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