A radio pulse is emitted from the spaceship towards Earth. In the reference frame of Earth, the emission occurs at a distance of $2\mathrm{.}00{10}^{6}m$ from Earth. The emission of the pulse is event $P$ and its detection is event $Q.$

b$.$ Construct events $P$ and $Q$ on the spacetime diagram.

c$.$ Calculate:

i$.$ the proper time interval between the launch of the spaceship and event $P$

ii$.$ the coordinates $(c{t}^{\text{'}},x^{\text{'}})$ of event $Q$ in the reference frame of the spaceship.

d$.$ State the distance travelled by the radio pulse according to:

i$.$ a spaceship observer

ii$.$ an observer on Earth.

$13.$ A spaceship is sent from Earth towards a distant planet. The diagram shows the spacetime axes $(x,ct)$ of the reference frame of Earth and the worldline ct of the spaceship. The spaceship leaves Earth at $t=0$

and moves at a constant velocity relative to $E_{\text{a}\text{r}\text{t}\text{h}}$ the direction of the negative $x-$axis. Coordinate, in are scaled in light years $($ly$).$ When $t=2\mathrm{.}00y_{\text{a}\text{r}\text{s}}$ communication signal travelling at the speed of$,$ A sent from Earth towards the spaceship.

a$.$ Copy the spacetime diagram and draw on it:

i$.$ the space axis $x$ for the reference frame of $f_h$ spaceship

ii$.$ the worldline of the communication signa$|.$

b$.$ Estimate, using the diagram, the time at which the communication signal is received:

i$.$ according to the clock in the spaceship

ii$.$ according to the clock on Earth.

c$.$ The speed of the spaceship is $0\mathrm{.}500$ c$.$ Determine using the Lorentz transformation, the spacetime coordinates in the reference frame of the spaceship of the event when the communication signal is received.