Three little asteroids, all in a row?

[bonanova]

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.

[CaptainEd]

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:

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 t₁ to 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.
 

 

[bonanova]

Right, CaptainEd, middle does not imply equidistant, and your answer is correct..