The center of mass of two bodies is determined by the difference in their inertial mass. If the two bodies have the same inertial mass then this point will be one half the distance between them. Or more specifically, the distance of M(adsbygoogle = window.adsbygoogle || []).push({}); _{1}to the center of mass is ##D\frac{M_{2}}{M_{1}+M_{2}}## and the distance of M_{2}to the center of mass is ##D\frac{M_{1}}{M_{1}+M_{2}}##, where D is the distance between the two bodies. So if the two bodies are on a collision course then this is the point where they will meet (barring their physical size by assuming test masses). If the two bodies are in orbit about each other then this is the point they will orbit around.

However, the acceleration, the time to impact, and the period of the orbit, are all determined by another property of mass. And that property is called the active gravitational mass. A measure of this property is called the standard gravitational parameter, usually denoted as ##\mu##. The acceleration of two bodies on a collision course with each other is determined by this property, or more specifically ##\frac{\mu_{1}+\mu_{2}}{R^2}##, where R is the distance between the two bodies.

Now, even though these two properties are completely different and have different units of measure they are proportionally equivalent. See the Equivalence Principle. So if you know the value of one, you can calculate the value of the other by using their factor of proportionality, which is the universal gravitational constant, or big G.

One of the main sticking points in understanding the universality of free fall, or UFF, is using an improper frame of reference. The UFF is only valid when using the center of mass as the frame of reference. However, notice in the second paragraph, which describes the effect that ##\mu## has on the acceleration, that the frame of reference is not the center of mass. The acceleration in this frame is called the relative acceleration and will not work for the UFF. It is the acceleration as viewed from one body to the other. To get the proper frame of reference we can use the method from paragraph one:

$$A_{1}=\frac{\mu_{1}+\mu_{2}}{R^2}\frac{M_{2}}{M_{1}+M_{2}}=\frac{\mu_{1}+\mu_{2}}{M_{1}+M_{2}}\frac{M_{2}}{R^2}=G\frac{M_{2}}{R^2}$$

$$A_{2}=-\frac{\mu_{1}+\mu_{2}}{R^2}\frac{M_{1}}{M_{1}+M_{2}}=-\frac{\mu_{1}+\mu_{2}}{M_{1}+M_{2}}\frac{M_{1}}{R^2}=-G\frac{M_{1}}{R^2}$$

The UFF is always true regardless of the difference in mass of the bodies.

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# B Why do objects fall at the same speed in free fall?

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