Gravitational Acceleration of
Two Point Masses

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Consider two point objects (i.e. particles) of mass m₁ and m₂ located on the x-axis at x₁ and x₂ respectively as shown in Fig. 1 below 

Let F₁ be the force on particle 1 due to particle 2, and let F₂ be the force on particle 2 due to particle 1. Then

Multiply Eq. (1a) by m₂ and Eq. (1b) by m₁ to give

Subtract Eq. (2b) from Eq. (2a) to give 

Eq. (3) simplifies to   

Cancel m₁m₂ and Eq. (1b) by m₁ to give R = x₂ - x₁ to give

Eq. (5) states that the relative acceleration of two point particles due to their mutual gravitational attraction is the same as the acceleration of a test particle in the gravitational field of and object whose mass is the sum of the two particles.

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