Two rods having the same proper length `l_0` move lengthwise toward each other parallel to a common axis with the same velocity v relative to the laboratory frame of reference. What is the length of each rod in the reference frame fixed to the other rod?

Explanation

The velocity of one of the rods in the reference frame fixed to the other rod is
`V=(v+v)/(1+v^2/c^2)=(2v)/(1+beta^2)`
The length of the moving rod in this frame is
`l=l_0sqrt(1-(4v^2//c^2)/((1+beta^2)^2))=l_0(1-beta^2)/(1+beta^2)`