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Relativistic Mechanics

Problem 1.352

A rod \(A B\) oriented along the \(x\) axis of the reference frame \(K\) moves in the positive direction of the \(x\) axis with a constant velocity v. The point \(A\) is the forward end of the rod, and the point \(B\) its rear end. Find: (a) the proper length of the rod, if at the moment \(t_{A}\) the coordinate of the point \(A\) is equal to \(x_{A},\) and at the moment \(t_{B}\) the coordinate of the point \(B\) is equal to \(x_{B}\); (b) what time interval should separate the markings of coordinates of the rod's ends in the frame \(K\) for the difference of coordinates to become equal to the proper length of the rod.

Reveal Answer
 (a) l0=xA−xB−v(tA−tB)1−(v/c)2 (b) tA−tB=(1−1−(v/c)2)l0/v or tB−tA=(1+1−(v/c)2)l0/v\text { (a) } l_{0}=\frac{x_{A}-x_{B}-v\left(t_{A}-t_{B}\right)}{\sqrt{1-(v / c)^{2}}} \text { (b) } t_{A}-t_{B}=\left(1-\sqrt{1-(v / c)^{2}}\right) l_{0} / v \text { or } t_{B}-t_{A}=\left(1+\sqrt{1-(v / c)^{2}}\right) l_{0} / v
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