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 $\hat{\Delta} t_{0}$