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Just wanted to check if I've done this right, because it isn't working.

Mercury: mass $$3.285 \times 10^23\ \mathrm{kg}$$ position: $57,910,000,000\ \mathrm{m}$ from sun

Earth: mass $$5.972 \times 10^24\ \mathrm{kg}$$ position: $149,600,000,000\ \mathrm{m}$ from sun

Sun: mass $$1.98855 \times 10^30\ \mathrm{kg}$$ position: start it off at $(0, 0)$. Assume all y co-ordinates are zero at start. To work out where the center of mass is, use $$Mx_{\text{c.o.m.}} = \sum mx$$ rearrange for $$x_{\text{com}}$$ and compute to get the center of mass at $x = 458844$. To be in the center of mass frame, I now have to subtract this from the initial positions.

Sun: position $(-458844, 0)$

Earth: $149,995,000,000$

Mercury: $57,909,500,000$,

To work out their velocities about the center of mass, use $v = \large \frac{2\pi r}{T}$, and here's the problem: What is $T$ for mercury, the Earth and the Sun? Earth around Sun is $365$ days, Mercury round Sun is $88$ days, but what is their period around the center of mass?

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Well, the title of your question says that you need to find the velocity of center of mass, but, your description says you need to know the velocity of each planet around this center of mass.

If you mean the velocity of center of mass then simply take the derivative of $x_{\text{c.o.m}}$ and put the velocities of each planet in the formula and you should be good:

$$V=\frac{1}{M}\sum m_iV_i$$

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  • $\begingroup$ But to work out the velocity of the centre of mass, I need the velocity of planets about that centre of mass, don't I? And I can't work out their velocities unless I know their period of orbit around the centre of mass. $\endgroup$ – user13948 Dec 5 '15 at 10:16
  • $\begingroup$ If you calculate in a coordinate system that is attached to a part of space, say sun, then the velocity of center of mass is calculated by the formula above, where the velocity of each planet is with respect to the coordinate system, here attached to the sun. Then, NO, you do not need to know the relative speed of planets with c.o.m, BUT, remember to consider the vector nature of your calculation. $\endgroup$ – Amin R. Dec 5 '15 at 10:45

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