All Problems

Relativistic Mechanics

Problem 1.362

An unstable particle moves in the reference frame \(K^{\prime}\) along its \(y^{\prime}\) axis with a velocity \(v^{\prime}\). In its turn, the frame \(K^{\prime}\) moves relative to the frame \(K\) in the positive direction of its \(x\) axis with a velocity \(V\). The \(x^{\prime}\) and \(x\) axes of the two reference frames coincide, the \(y^{\prime}\) and \(y\) axes are parallel. Find the distance which the particle traverses in the frame \(K,\) if its proper lifetime is equal to \(\hat{\Delta} t_{0}\)

Reveal Answer
s=Δt0V2+(1−β2)v′2(1−β2)(1−v′2/c2), where β=V/cs=\Delta t_{0} \sqrt{\frac{V^{2}+\left(1-\beta^{2}\right) v^{\prime 2}}{\left(1-\beta^{2}\right)\left(1-v^{\prime 2} / c^{2}\right)}}, \text { where } \beta=V / c
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