A space colony defines the origin of a coordinate system in reference frame \(\mathrm{C}\), and you are at rest in this reference frame at position \(x=+3.000 \times 10^{5} \mathrm{~km}\). A ship passes the colony moving in the positive \(x\) direction at \(0.482 c_{0}\) in reference frame \(\mathrm{C}\). By prearrangement, observers on the ship (reference frame S) and on the colony use this event (event 0 ) to define \(x=0\) and \(t=0\) in both reference frames. Just as the ship passes, the colonists send out a light pulse toward you (event 1), and you transmit a reply pulse (event 2 ) immediately upon receipt.

(a) At what position in reference frame \(S\) do observers see the colonists' pulse originate?

(b) At what instant does the pulse originate according to these observers?

(c) At what position in reference frame \(S\) do the observers see your reply pulse originate?

(d) At what instant does the reply pulse originate according to observers on the ship?