Given the point mass $p_1$ at $(0,0)$ with mass $m_1$ and the point mass at $p_2$ at $(r,0)$ with mass $m_2$ how would you find the positions of $p_1$ or $p_2$ at any time. My first thought was to first solve the problem of what their positions were after some small period of time $\Delta t$. The force by gravity on $p_1$ is $F_1=\frac{Gm_1m_2}{r^2}$ and from the equation $F=ma$ substituting $\frac{v}{t}$ in for $a$ and the $\frac{d}{t}$ for $v$ the result should be $F=\frac{md}{t^2}$ we know the force on $p_1$ and its mass, as well as the time so substituting them results in $\frac{Gm_1m_2}{r^2}=\frac{m_1 d}{\Delta t^2}$ some quick algabra tells me that $p_1$ should have move a distance of $d_1=\frac{Gm_2\Delta t^2}{r^2}$. Taking similar steps brings me to find that $p_2$ travels a distance of $d_2=-\frac{Gm_1\Delta t^2}{r^2}$. From there the same process could be repeated to find the position of of $p_1$ and $p_2$ at the time $2\Delta t$ except by using $r-\frac{G(m_1+m_2)\Delta t^2}{r^2}$ for the new distance between the points. In general if $R(t)$ is the distance between $p_1$ and $p_2$ and if $D_1(t)$ is the distance that $p_1$ moves over the time $\Delta t$ at time the time $t$ then $D_1(t)=\frac{Gm_2\Delta t^2}{R(t)^2}$ and the corresponding function $p_2$ is $D_2(t)=\frac{Gm_1\Delta t^2}{R(t)^2}$. With those two equations $R(t)=\lim_{h\to\infty }\sum_{n=0}^h D_1(\frac{nt}{h})+D_2(\frac{nt}{h})$, but since $D_1$ and $D_2$ are defined that makes are self referential, is there any way out of that? Would the definition $R(t)=\int_0^t\frac{G(m_1+m_2)}{R(u)^2}du$ be equivalent or useful? Most of this work is back of the envelope stuff combined with only really a half understanding of both calculus and Newtonian Mechanics, so any pointers or advice would be greatly appreciated.
Asked
Active
Viewed 94 times
-2
-
I think that you need to slow down and work out the equations more carefully because to me it seems like you're speeding through your back-of-the-envelope calculations while neglecting important details. One important thing that you left out is velocity. You seem to be assuming that both points start out with zero velocity (OK, but you should explicitly note that). Also, in the subsequent time steps you seem to be neglecting the fact that the points will have non-zero velocities since your displacements are proportional to $(\Delta t)^2$ whereas there should also be a $\Delta t$ term in them. – Jun 15 '18 at 20:13
-
You're right, I completely forgot about that. Is there a better way to start off, or a specific path that I should be taking instead of speeding through? – Aaron Quitta Jun 15 '18 at 20:24
-
Just work through the equations carefully step by step. Also, you probably want to have some initial velocities and do the problem in 2D or 3D because otherwise the particles will crash directly through each other and and you'll have the gravitational force between them shooting up to infinity as the distance between them approaches zero. – Jun 15 '18 at 20:50
-
Which equations do I need to work through? I still am unsure on which ones are correct. The fact that they will adventually collide is unavoidable in a two body point system, and I would think that the introduction or radi would only further complicate things. The relevance of my math in real life is of secondary concern. – Aaron Quitta Jun 15 '18 at 21:11
-
You may need an equation for predicting velocity (as one comment pointed out). This can also be derived from Newton's laws, if you solved for (x,y,z) you should be able to find its derivative. then the question is how meany independent P measurements or observations are needed to completely specify the unknown quantities and start predicting. – Jun 16 '18 at 01:09
-
This question is very hard to read, it looks like a jumble. If you edit some of the equations into blocks, for example with the double-dollar-sign latex notation it might make this easier to read and you will get better answers. – hft Jun 26 '18 at 04:09
1 Answers
0
Given the point mass p1 at (0,0) with mass m1 and the point mass at p2 at (r,0) with mass m2 how would you find the positions of p1 or p2 at any time.
This is a difficult problem unless you use a well-known trick: break the problem into the motion of the center-of-mass and the relative motion.
The center-of-mass ($(m_1\vec r_1 + m_2\vec r_2)/(m_1+m_2)$) obeys a very simple equation...
The relative coordinate ($\vec r_1 - \vec r_2$) obeys an equation that you can solve using basic calculus.
$$ \left( \frac{m_1 m_2}{m_1+m_2}\right)\frac{d(\vec r_1 - \vec r_2)}{dt} = -\frac{Gm_1 m_2 (\vec r_1 - \vec r_2)}{|\vec r_1 - \vec r_2|^3} $$
hft
- 19,536
-
I'm assuming $r_1^\rightarrow$ and $r_2^\rightarrow$ are the distances from $m_1$ and $m_2$ to the center of mass respectively? – Aaron Quitta Jun 26 '18 at 04:31
-
Not according to what you have written in your original post, you're not... You have not stated anything about center-of-mass in your original post. You've only written "Given the point mass p1 at (0,0) with mass m1 and the point mass at p2 at (r,0) with mass m2." In which case the center-of-mass is initially located at $(m_2/(m_1+m_2))(r,0)$. – hft Jun 26 '18 at 04:34
-
To clarify, I have no idea what those symbols represent, I was guesing and asking you. – Aaron Quitta Jun 26 '18 at 04:36
-
oh... no, the symbol $\vec r_1$ is the vector giving the location of point p_1, which is (0,0) initially. The symbol $\vec r_2$ is the vector giving the location of point p_2, which is (r,0) initially. – hft Jun 26 '18 at 04:37
-
-
The vector $\vec r_1$ is defined as the the vector specifying the location of point mass $p_1$. The vector $\vec r_2$ is defined as the vector specifying the location of point mass $p_2$. – hft Jun 26 '18 at 04:43
-
I see. I only have limited knowledge when it comes to subtracting vectors and even less on cubing not to mention doing derivatives with them, is there a resource you can point me to for help? – Aaron Quitta Jun 26 '18 at 04:47
-
Any basic physics textbook (for example, "Fundamentals of Physics" by Halliday, Resnick, and Walker). Any textbook on classical mechanics (for example "The Theory of Classical Dynamics" by J. B. Griffiths) – hft Jun 26 '18 at 04:51