An unstable particle moves in the reference frame `K^'` along its `y^'` axis with a velocity `v^'`. In its turn, the frame `K^'` moves relative to the frame K in the positive direction of its x axis with a velocity `V`. The `x^'` and x axes of the two reference frames coincide, the `y^'` and `y` axes are parallel. Find the distance which the particle traverses in the frame K, if its proper lifetime is equal to `Deltat_0`.

Explanation

The components of the velocity of the unstable particle in the frame K are
`(V, v^'sqrt(1-V^2/c^2),0)`
so the velocity relative to K is
`sqrt(V^2+v^('^2)-(v^('^2V^2)/(c^2))`
The life time in this frame dilates to
`Deltat_0//sqrt(1-V^2/c^2-v^('^2)/c^2+(v^('^2)V^2)/(c^4))`
and the distance traversed is
`Deltat_0(sqrt(V^2+v^('^2)-(v^('^2)V^2)//c^2))/(sqrt(1-V^2//c^2)sqrt(1-v^('^2)//c^2))`