A problem regarding time dilation

[THE HARLEQUIN]

two spacecraft s travel parallel to each other along straight lines AB and CD as shown in the picture here ...
while s1 moves with a simple harmonic motion , s2 moves with a constant speed and only at points A and B it gets enough thrust within a very negligible time to reverse its direction of velocity ...
s1 and s2 travel the same AB=CD distance after a same amount of time ..
now both of the spacecraft s fire rockets R1 and R2 at the same time from points D and B respectively and they come back after time t .
now if S1,R1, S2,R2 carry 4 persons of the same age then after time t when they meet
1. what will be the difference between their ages ?
2. on what condition it is possible for all of them to have the same age when they meet ?
[ note: you can take v of S2 and f of S1 arbitrarily ]
answers with proper visualizing power will be appreciated ... mathematical analysis is also appreciated but not needed ...

THE HARLEQUIN

 

[Nugatory]

Draw a space-time diagram. Spaceship A's world line will be a sinusoidal curve, except oriented vertically instead of horizontally because the t axis is vertical. Spacecraft B's world line will be a zigzag line. The worldlines of the two rockets will be straight line segments, starting at the event where they're launched. The change of age of each person is given by the spacetime distance along their worldline between the starting point and the end point.

That's your visualization tool. To actually calculate the changes in age, you'll need to do a (straightforward) line integral to calculate the change of age along the sinusoidal world line. For the straight segments, we don't need the calculus - ##\Delta{\tau}=\sqrt{\Delta{t}^2-\Delta{x}^2}## will do the trick.

 

[THE HARLEQUIN]

Draw a space-time diagram. Spaceship A's world line will be a sinusoidal curve, except oriented vertically instead of horizontally because the t axis is vertical. Spacecraft B's world line will be a zigzag line. The worldlines of the two rockets will be straight line segments, starting at the event where they're launched. The change of age of each person is given by the spacetime distance along their worldline between the starting point and the end point.

That's your visualization tool. To actually calculate the changes in age, you'll need to do a (straightforward) line integral to calculate the change of age along the sinusoidal world line. For the straight segments, we don't need the calculus - ##\Delta{\tau}=\sqrt{\Delta{t}^2-\Delta{x}^2}## will do the trick.

thanks .. that was helpful ... i also want to know if the spacecraft 2 has the same velocity as the rocket will the age of the man in in the spacecraft and the man in the rocket differ ? if it does then by what factor ?

 

[Nugatory]

Draw your spacetime diagram, label the end points of the relevant line segments, calculate the amount of aging on each path... And let us know what you find.