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Three little asteroids, all in a row?

Question

Three asteroids are traveling in a gravity-less region of space. At time t = t1 they occupy collinear positions. At a later time t = t2 their positions are also collinear. The asteroid in the middle at t1 is also in the middle at t2.  Will the asteroids always be collinear? Assume the asteroids are light enough, and distant enough, that their mutual gravitational attraction is negligible.

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Suppose the point in the middle is called C . By "in the middle", do you mean "C is equidistant from the other two points"? or do you simply mean a line segment from one point to the other passes through C? I imagine different answers for the two cases.

 

Assuming C is between A and B,

Spoiler

In general, C will not have to be collinear with the other two points. Here is an example picture:

Three Points.PNG

First, since all motions are linear, subtract the motion of C from all three points, and display C as stationary.

In the diagram, one point moves from A at tto A1 at t2, the other moves from B at t1 to B' at t2
project each path forward the same amount, so that the points reside at A'' and B'' at time t3.,

The blue dotted lines represent the collinearity at t1 and t2.
The red dotted line connects A'' and B'', and clearly does not include point C.
 

 

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