Rate of Change of the Distance between Two Point Masses

The green curve is the parametric plot of the path and the blue curve is the parametric plot of the path . The point masses and move along the green and blue curves, respectively, as the time varies. As we can see, both paths are sinusoidal functions with chosen amplitudes , and frequencies , . The Demonstration keeps track of the distance between the two point masses, and the rate of change of this distance with respect to time , that is, the time derivative of the distance.

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The paths of the two point masses, for a given choice of frequencies (for example, whenever and ), can model two satellites orbiting elliptically around some massive body. As a special case, if or , one or both point masses move in a circular orbit. A modern GPS receiver calculates its position by measuring its distance from GPS satellites and is able to measure the distance between the satellites.
The paths of the point masses for other choices of frequencies are Lissajous figures. The pattern of a Lissajous curve is highly sensitive to the ratio of the frequencies , . Ratios other than give rise to complicated Lissajous patterns. When the ratio is rational, the curve is closed.
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