Suppose two and only two planets with point masses are colliding due to gravity. I am trying to solve for
- Final displacement of the two planets
- Time-dependent displacement functions of the two planets
By Newton's Gravitational Law, $\displaystyle F=\frac{Gm_1m_2}{r^2}$ where $r$ is the distance between the two positions of the planet. The core problem is that $F$ or the acceleration increases as distance $r$ decreases.
Let the following variables:
- $m_1,m_2$: mass of the two planets respectively
- $s_{f1},s_{f2}$: final displacements of the two planets respectively
- $s_1(t), s_2(t)$: displacement functions of the two planets respectively
- $r_0$: initial distance of the two planets
- $r(t)$: time-dependent total distance between the planets
Is the ratio of displacements of the two planets always negatively equal to the inverse ratio of mass of the two planets? Here is what I did: ($u$ is a dummy variable) $$m_1\frac{d^2s_1}{dt^2}=-m_2\frac{d^2s_2}{dt^2}$$ $$m_1\int^t_0\int^u_0\Big(\frac{d^2s_1}{dt^2}\Big)dt=-m_2\int^t_0\int^u_0\Big(\frac{d^2s_2}{dt^2}\Big)dt$$ $$m_1s_1\Big|^t_0=-m_2s_2\Big|^t_0$$ $$\frac{m_1}{m_2}=-\frac{s_2\Big|^t_0}{s_1\Big|^t_0}$$ If that is the case then $\displaystyle s_{f1}=r_0\frac{m_2}{m_1+m_2}$ and $\displaystyle s_{f2}=r_0\frac{m_1}{m_1+m_2}$
How do I find $s_1(t)$ and $s_2(t)$ analytically? I currently can think of two separate approaches:
$$\frac{\partial ^2s_1}{\partial t^2}=G\frac{m_2}{(r_0-s_1-s_2)^2}$$ $$\frac{\partial ^2s_2}{\partial t^2}=G\frac{m_1}{(r_0-s_1-s_2)^2}$$
$$\frac{d^2r}{dt^2}=\frac{G(m_1+m_2)}{(r_0-r)^2}$$ and then use the displacement ratio proved above to find the respective displacement functions.
How do I solve the differential equations analytically?
