$A_{1}$ moves at speed $v_{1}$ in the $x$-direction with respect to $A_{0} . A_{2}$ moves at speed $v_{2}$ in the $x$-direction with respect to $A_{1}$. By applying a Lorentz transformation between the rest frames of $A_{0}, A_{1}$, and $A_{2}$, calculate the speed at which $A_{0}$ observes $A_{2}$ to travel.

$A_{3}$ moves at speed $v_{3}$ in the $x$-direction with respect to $A_{2}$. Calculate the speed at which $A_{0}$ observes $A_{3}$ to travel.