Confused about Special Relativity

[EngageEngage]

Homework Statement

Suppose that we send a spaceship on a trip to Alpha Centauri, 4 light years away. The spaceship travels at a constant speed of .5c, it stops at Alpha C. for 1 year and then it returns to the Earth at constant speed .5c.
a. Draw a spacetime Diagram with axes x,ct and plot the worldine of the spaceship.
b. suppose that 1 year after the departure of the spaceship we send a radio message from Earth to the spaceship. Plot the worldline of the radio signal.
c. At what time does it reach the spaceship?

Homework Equations

The Attempt at a Solution

a.
ct
|\
| .\
| .. \
| ... |
| .. /
| ./
|/________________x
where the angle between the ct and x-axis is given by

theta = arctan(.5)

b.)I drew a cone with an angle of 45 between the ct and x axis.
c.) this is what I'm confused about. I'm not sure if I have to take relativity into account since c is the same in all reference frames. So, my answer is 1 year after the message is sent.

Can anyone please tell me if any of this is right? Any help will be greatly appreciated.

 

[EngageEngage]

does anyone have any ideas? I'm still really confused about this one. The only things that we've gone over so far are the basic Gallilean transformations and Lorentz transformations. But, i just don't see how either of them would help here. Once again, if anyone could offer any guidance or feedback I would really appreciate it.

 

[Moetasim]

Your diagram for part a looks correct, with the angle between ct and x-axis being arctan(0.5) = 26.6 degrees.

For part b, the radio signal will also travel at the speed of light, so it will have a worldline that is a straight line at a 45 degree angle from the ct-axis. This means that the radio signal will reach the spaceship at the same time that it was sent, which is 1 year after the spaceship's departure.

In special relativity, time is relative and depends on the observer's frame of reference. However, the speed of light is always constant in all frames of reference. This means that the time it takes for the radio signal to reach the spaceship will be the same in both the Earth's frame of reference and the spaceship's frame of reference. Therefore, 1 year after the spaceship's departure is the correct answer.

I hope this helps clarify any confusion you may have had about special relativity. Keep practicing and you will become more comfortable with it over time.