Radial Motion of Two Mutually Attracting Particles

摘要

A pair of masses or opposite-sign charges released from rest will move directly toward each other under the action of the inverse-distance- squared force of attraction between them. An exact expression for the separation distance as a function of time can only be found by numerically inverting the solution of a differential equation. A simpler, approximate formula can be obtained by combining dimensional analysis, Kepler's third law, and the familiar quadratic dependence of distance on time for a mass falling near Earth's surface. These exact and approximate results are applied to several interesting examples: the flight time and maximum altitude attained by an object fired straight upward from Earth's surface; the time required for an asteroid of known starting position and speed to cross Earth's orbit if it is bearing toward the Sun; and the collision time of two oppositely charged particles starting from rest.