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Imagine you have two homogeneous spheres with the same diameter of $d=0.1 m$. They have the same mass $m = 1 kg$. The distance between the centers of mass is $r= 1 m$. Their electrical charge shall be disregarded. At $t=0$ the spheres do not have any relative motion to each other. Due to gravitation they will accelerate and start moving towards each other. After some time they will touch each other.

How to calculate analytically the time it takes the two spheres to meet each other. I'm not interested in a numerical solution.

I have already tried several ways but I don't get to a solution.

Imagine that the 2 spheres have different masses and diameters. $m_{1}=2 kg$, $m_{2}=5 kg$, $d_{1}=0.03 m$, and $d_{2}=0.3m$. How to calculate analytically when and where the 2 spheres are going to meet?

How do you calculate the second problem taking the theory of relativity into account? I know that it will not change the result that much but I am interested in the mathematical solution.

Sensebe
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user50224
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1 Answers1

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Qmechanic basically answered it in a comment, but to reiterate, the Wikipedia article on free-fall details this exact scenario: the time as a function of separation becomes $$t(y)=\sqrt{\frac{y_0^3}{2G(m_1+m_2)}}\left(\sqrt{\frac{y}{y_0}\left(1-\frac{y}{y_0}\right)}+\cos^{-1}\left(\frac{y}{y_0}\right)\right)$$ where $y_0$ is the initial separation. Letting the radii be $r_1,r_2$, this yields a collision time $$T=t(r_1+r_2).$$

DumpsterDoofus
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