# Special Relatvity & Spacetime Diagrams

1. Oct 1, 2008

### shukuchi

1. The problem statement, all variables and given/known data
Hansel and Gretel both make a journey from Earth to the Gingerbread Planet, 1 light year away. Hansel travels at a constant speed of 0.3c there and back. Gretel leaves Earth at the same time, but travels at a constant acceleration (in the Earth's frame), and chooses this acceleration and her initial velocity so that she and Hansel reach the Gingerbread Planet simultaneously and arrive back at Earth simultaneously.

a) What is Gretel's acceleration and her initial velocity (no relativity required for this part). Hint: her acceleration is negative, and her velocity should be zero when she reaches the Gingerbread Planet

b) Sketch Hansel and Gretel's trajectories on a spacetime diagram

c) If Hansel and Gretel are the same age when they leave the Earth, who is older when they return, and by how much?

2. Relevant equations
$$\gamma$$ = 1 / $$\sqrt{(1 - v^2/C^2)}$$

3. The attempt at a solution

a) Since the events (Hansel and Gretel travelling to the planet and back) are simultaneous, the two time intervals should be the same in the Earth's frame.

The time it takes Hansel is : t(Hansel) = D / v = 1*c year / 0.3c = 3.333 years. I think we might need to take into account time dilation, but the questions says no relativity required for part a....

So I plan on using the simple formula v(final) = v(initial) + at(Gretel) for Gretel, where t(Hansel) = t(Gretel). But the thing is I have two variables but only one equation to solve. I'm stuck.

b) I would assume the position and time when they are at Earth to be t = 0, x = 0. The direction they are travelling towards the planet is the +x direction, and conversely the direction the are travelling back is -x.

I don't know how to paste my sketch online, so I'll give a description of the graph.
The y-axis is labelled ct and the x-axis is labelled x.

Hansel's trajectory will have slope of c/v = c / 0.3c = 3.333 from x = 0 to x = 1. After that it will turn back towards Earth, so we should expect the trajectory to curve backwards toward x = 0. So it will look like kinda "v-shaped" sideways.

Since I don't have the information for Gretel for part a, I don't know how to graph it but the two graphs should have the same slopes because they are simultaneous right?

c) I would probably need to know a and b in order to know this. My guess would be Gretel. If her ship has accelerated to a speed greater than 0.3c, she would experience a greater time dilation effect.

Sorry guys, this is my first post on this forum, I'm just beginning to learn these stuff and I'm having GREAT difficulty grasping the concepts. I know my attempted solutions aren't very mathematical and incomplete, but I'm really unsure how to continue this question.