So say there are two point masses, m1 and m2. There is no gravity affecting them besides the gravity between themselves. How long does it take them to collide with a given distance? The acceleration will change at the rate of x^2, so the objects will experience constant jounce, the second derivative of acceleration. There are a couple problems i am having with solving this equation. The first is how to find the acceleration toward the other obejct of each individual mass. If i had the initial acceleration, I could make that a function of acceleration vs time, and making a differential equation solve for distance vs time. With that, all I would have to do is find out how far the objects have to travel to reach each other, which leads into my second problem. How can i find where the objects will collide? It makes sense that they will meet at their inital barycenter, but I am not completely sure.
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As for your second question, use, they have to meet at the barycenter. That's by conservation of momentum. The barycenter has zero initial velocity, so it has to stay in the same place. – knzhou Sep 05 '16 at 22:49
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The energy of a system of two masses $M~>>~m$ is $$ E~=~\frac{1}{2}mv^2~-~\frac{GMm}{r}. $$ If we start the particle with mass $m$ at a distance $R$ with zero velocity then $E~=~-\frac{GMm}{R}$ is constant. Now let $v~=~dr/dt$ for a straight radial infall so that, $$ -\frac{GMm}{R}~=~\frac{1}{2}m\left(\frac{dr}{dt}\right)^2~-~\frac{GMm}{r}. $$ Now algebraically rearrange this and integrate $$ \int dt~=~T~=~\int_R^{r'}\frac{dr}{\sqrt{2GM(\frac{1}{r}~-~\frac{1}{R})}}. $$ The integral can be solved with MATHEMATICA etc.
For the two masses arbitrary one can work with the reduced mass $\mu~=~m_1m_2/(m_1+m_2)$. The result will be similar.
Lawrence B. Crowell
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