Gravity paths
Problem
An object of mass $m$, perhaps a spaceman, is dropped from rest in space and accelerates under the influence of a series of stationary planets.
In each of the following seven scenarios, the red planets have the same mass $M$ and lie with their centres on grid axes. Assume that all of the masses are in a plane, so that the motion is entirely two-dimensional.
For the indicated placement of the object (black circle), in which cases can you describe for certain the subsequent qualitative form of the motion? In which cases are there a variety of possible motion types, depending on the magnitudes of $m$ and $M$? Where the form of the motion is unclear, you might be able to use Newton's law of gravitation to help
resolve the uncertainties.
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Use vectors to determine in each case the exact direction in which the object will start to fall.
Discussion: in each diagram, can you locate a set of initial points for which the object would never fall into one of the planets? Can you be sure that you have found all such points?