In a triangle the proper length of each side equals a. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity V along one of its bisectors :
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Solution
Length of the rod moving parallel to the direction of velocity (v) gets reduced by a factor of √1−β2 of its proper length whereas the length perpendicular to the direction of velocity remains the same.
Given : |AB|=|AC|=|BC|=a (proper length)
In the moving frame : A′B′=a
→AC=acos(30)^i+asin(30)^j
∴→A′C′=[(acos30)√1−β2^i+asin30^j]=a√3(1−β2)2^i+a2^j where β=v/c
⟹|A′C′|=a√4−3β22
Similarly |B′C′|=a√4−3β22
∴ Perimeter of the triangle P′=|A′B′|+|A′C′|+|B′C′|=a+a√4−3β22+a√4−3β22