What Is the Formula for the Position of a Mass Falling Towards a Planet?

[Zeeprime]

Question: Finding the closed formula s(t) that gives the approaching position of an inertial mass to a planet
Supposing the mass initially stationary, and far enough and for long enough that it is NOT possible to consider the gravity as constant while it moves closer and closer.

Said in a different way.
Given a mass in free fall from afar to a planet, what is the motion formula s(t),
the one that returns:
at 0 seconds the mass will be 100000km distant
at 10 seconds the mass will be 99999Km distant.
at 20 seconds the mass will be 99996Km distant
at 100 seconds the mass will be 94000Km distant (it is accelerating, and the acceleration increases while it approaches the planet), etc.

So given M= the mass of the planet
m = the mass of the free fall mass
So = the initial distance of the mass

Given F=G(Mm)/r^2
we obtain
m*a(t) = GMm/(r(t)^2)
a(t) = GM/(r(t)^2)

Now we solve with derivative calculus
Posing k=GM
s'' = k/(s^2)

Is the above correct?
If it is, how can I solve this differential equation?
(I suppose a constant will pop out from some integral, and it will be our initial S0)

Can I simply integrate left and right a couple of times?
Providing all is preserved, no bad negatives, no 0s around, all functions are analytical in complex space as they appear to be etc.

If I can, I suppose that I will end up with
s' = S0-k/s
s = S0-k/(ln(s))

which is:
s(t) = S0-k/(ln(s(t))

And now? How can I get s(t) ?

Another way to solve this problem?

 

[Buzz Bloom]

Hi Zeeprime:

The following may be helpful.

https://en.wikipedia.org/wiki/Radial_trajectory​

Regards,
Buzz

 

[Zeeprime]

Hi Zeeprime:

The following may be helpful.

https://en.wikipedia.org/wiki/Radial_trajectory​

Regards,
Buzz

Thanks Buzz.
I am really confused now.
I hoped this problem were easier, instead the closest thing I have found on your links is

https://en.wikipedia.org/wiki/Kepler's_equation#Radial_Kepler_equation

Which gives some sort of time in function of space

And the inverse formula does not look like being easy at all! :(

 

[John Park]

Let's go back to your equation
s'' = k/(s^2)

You tried to integrate this directly, giving s' on the left hand side. In other words you were integrating with respect to t. So you'd also have to integrate the right hand side with respect to t, not s as you tried to do. The real next step is to note that s'' = v(dv/ds), where v=s'. Then you can integrate both sides w.r.t. s. There's maybe a page of algebra to get to the final result, depending on how fluent you are, but it's not as intimidating as the general formulae on Wikipedia make it look.

Yes you get t as a function of s; you'll probably find it involves inverse trig functions. That's the way the world works. You can get the answers you want by simple interpolation, or by inverting the expression numerically, for instance using a Newton-Raphson algorithm.

(Note that you're considering a special case, where the mass is falling directly towards the centre of the planet, and has negligible mass compared to the planet.)

 

[Zeeprime]

Thanks.
I will try developing from your advice