Problem in Newtonian gravity- 2nd order, integration problems

In summary, the conversation discusses a problem involving finding the time for a body of mass m to fall into a sun of mass M, given a distance r from the sun. The conversation includes an initial equation and an attempted integration, as well as a solution provided by another user. The issue of double posting is also addressed.
  • #1
Romeo
13
0
The problem is this:

Given sun of mass M and a body of mass m (M>>m) a distance r from the sun, find the time for the body to 'fall' into the sun (initially ignoring the radius of the sun).


Our first equation is therefore [tex] \frac {d^2r}{dt^2} = \ddot{r} = \frac {GM}{r^2} [/tex].

I am able to integrate this, giving:
[tex] \dot{r} = - {\sqrt{2GM}}{\sqrt{1/r - 1/R}} [/tex],

where R is the inital distance of the body from the sun. However, I am unable to integrate this again. I have shoved it into wolfram's integrator for an indicator of what to aim for, but cannot come close.

Any thoughts would be greatly appreciated.

Regards

Romeo
 
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  • #2
This diff. eq. is separable.
We may write it as:
[tex]\sqrt{\frac{r}{R-r}}\frac{dr}{dt}=-\sqrt{\frac{2GM}{R}}[/tex]
Integrating from t=0 to t=T, where T is the time when r=0, we have:
[tex]\int_{0}^{R}\sqrt{\frac{r}{R-r}}dr=\sqrt{\frac{2GM}{R}}T[/tex]

Use the substitution [tex]u=\sqrt{\frac{r}{R-r}}[/tex] to progress further:
We get: [tex]r=R\frac{u^{2}}{1+u^{2}}, \frac{dr}{du}=2R\frac{u}{(1+u^{2})^{2}}[/tex]
And [tex]r=0\to{u}=0,r=R\to{u}=\infty[/tex]

Therefore, we have:
[tex]\int_{0}^{R}\sqrt{\frac{r}{R-r}}dr=2R\int_{0}^{\infty}\frac{u^{2}du}{(1+u^{2})^{2}}=R(arctan(u)-\frac{u}{1+u^{2}})\mid_{u=\infty}-R(arctan(u)-\frac{u}{1+u^{2}})|_{u=0}=\frac{\pi{R}}{2}[/tex]
That is:
[tex]T=\frac{\pi}{\sqrt{GM}}(\frac{R}{2})^{\frac{3}{2}}[/tex]
 
Last edited:
  • #3
Much appreciated Arildno and apologies if this is a little late coming. I hope my double posting was not too imposing- it would have been unnecessary had the original post in the College Homework forum taken a helpful direction.

Regards

Romeo
 
  • #4
No problem.
I see from your post count that you are fairly new here, so the rule about not double posting have naturally escaped you (you're not alone in this..:wink:).

Welcome to PF, BTW.
 
  • #5
Thanks Arildno. I was a little concerned about double (incidentely, it was a triple...) posting, but did so only because I thought patrons of the mathematics section may have a better insight- which seemed the case, since you very quickly responded :).

I'll keep it in mind for the future. Until then, if I have a problem that seems to be unresolved and decaying in one forum, is there any precedent for moving the thread to another forum, for fresh ideas?

Regards

Romeo
 
  • #6
1. First "bump" your post (i.e, write a new reply like "Hello? Culd I have some help here, please?")

2. If that doesn't work, and it is really critical, you might consider PM'ing the moderator for the forum.
 

Related to Problem in Newtonian gravity- 2nd order, integration problems

1. What is Newtonian gravity and why is it important?

Newtonian gravity is a fundamental law of physics that describes the gravitational force between two objects. It is important because it explains the motion of celestial bodies and plays a crucial role in understanding the dynamics of our universe.

2. What are 2nd order integration problems in Newtonian gravity?

2nd order integration problems in Newtonian gravity refer to situations where the equations of motion require the use of second-order differential equations to accurately describe the behavior of objects under the influence of gravity.

3. How are 2nd order integration problems solved in Newtonian gravity?

2nd order integration problems in Newtonian gravity can be solved using various mathematical techniques such as numerical integration, analytical solutions, or by using computer simulations.

4. What are the real-life applications of solving 2nd order integration problems in Newtonian gravity?

The solutions to 2nd order integration problems in Newtonian gravity have numerous applications in fields such as astronomy, space exploration, and engineering. For example, they are used to calculate the trajectories of spacecraft, satellites, and planets.

5. Are there any current challenges or limitations in solving 2nd order integration problems in Newtonian gravity?

Yes, one of the main challenges in solving 2nd order integration problems in Newtonian gravity is dealing with complex and nonlinear systems, which can be computationally intensive and require advanced numerical methods. Additionally, there are still some unresolved issues in understanding the behavior of objects in extreme gravitational conditions, such as near black holes.

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