A particle decays after traveling $600\text{m}$ in Earth's frame while moving at $0.80c$. What is the particle's lifetime in its own rest frame?

In Earth's inertial frame, two events are simultaneous and are separated by a distance of $1.80\times 10^8\text{m}$. What is the magnitude of the time difference between the events in a frame moving at $0.60c$ in the direction from the first event to the second?

In frame $S$, two events occur at the same place but are separated by $2.00\mu s$. What is the spatial separation between the events in a frame $S^{\prime }$ moving at $0.80c$ relative to $S$?

An unstable particle at rest has a proper lifetime of $2.5\mu \text{s}$. How fast must it travel so that only 1% of the particles in a beam survive after traveling $600\text{m}$?

A fixed monitoring station (frame $S$) detects two flashes produced by separate signal lamps. Event 1 takes place at the origin ($x_1=0,t_1=0$), while Event 2 takes place at $x_2=900\text{m}$ and $t_2=0$. A survey craft (frame $S^{\prime }$) moves in the positive $x$-direction with a constant speed of $0.80c$ relative to the monitoring station.