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Relativistic Mechanics

Problem 1.352

A rod ABA B oriented along the xx axis of the reference frame KK moves in the positive direction of the xx axis with a constant velocity v. The point AA is the forward end of the rod, and the point BB its rear end. Find: (a) the proper length of the rod, if at the moment tAt_{A} the coordinate of the point AA is equal to xA,x_{A}, and at the moment tBt_{B} the coordinate of the point BB is equal to xBx_{B}; (b) what time interval should separate the markings of coordinates of the rod's ends in the frame KK 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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