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**1. Homework Statement**

Two particles (masses

*m*

_{1},

*m*

_{2}) are released from rest a distance

*D*apart in space. How long until they collide?

**2. Homework Equations**

The force between the particles is [itex]F_G(t)=\frac{Gm_1m_2}{r(t)^2}[/itex].

The center of mass is located a distance [itex]r_1(t)=\frac{m_2}{m_1+m_2}r(t)[/itex] from particle 1 and [itex]r_2(t)=\frac{m_1}{m_1+m_2}r(t)[/itex] from particle 2. Note that [itex]r_1(t)+r_2(t)=r(t)[/itex].

**3. The Attempt at a Solution**

The collision will take place at the center of mass. Particle 1 needs to traverse a total distance of [itex]d_1=\frac{m_2}{m_1+m_2}D[/itex] to reach the COM. Its acceleration over this distance is

[itex]\ddot r_1(t)=\frac{m_2}{m_1+m_2}\ddot r(t)=\frac{Gm_2}{r(t)^2}[/itex].

OR! Another approach I thought of was to use potential and kinetic energy:

[itex]-\frac{Gm_1m_2}{D}=-\frac{Gm_1m_2}{r(t)}+\frac{1}{2}m_1v_1(t)^2+\frac{1}{2}m_2v_2(t)^2[/itex]

[itex]=-\frac{Gm_1m_2}{r(t)}+\frac{1}{2}m_1m_2\dot r(t)^2[/itex].

[itex]=-\frac{Gm_1m_2}{r(t)}+\frac{1}{2}m_1m_2\dot r(t)^2[/itex].

But either way, I wind up with a differential equation that's totally inhumane. I feel like I'm missing something here. Open to suggestion on how to further either of these two approaches, or other approaches entirely.