Observer O detects a sound pulse emitted at event A$.$ O observes the pulse to be reflected back in the opposite direction by a rigid barrier at event B$.$ The pulse is detected again at event C$.$ The space$-$time coordinates of the events according to O are:

A: $($xA $=0,$ tA $=0),$

B: $($xB$=+100$m$,$ tB$),$ and

C: $($xC $=0,$ tC $=2$tB$).$

 

In the following, three decimal places are sufficient,

$($a$)$ If the pulse travels at a constant speed $330$ m$/$s according to O$,$ what is tB$?$ Why?

$($b$)$ Observer O travels along the common x$,$ x direction with xA$=0$ at tA$=0.$ The velocity of O relative to O is $50$ m$/$s$.$ What are $($xB$,$ tB$)$ and $($xC$,$ tC$)$$?$ Why?

 

 $($c$)$ What is the velocity of the pulse between A and B and between B and C according to O$$$?$ Why?