# Twin Paradox from Moving Reference Frame- Return Journey

## Homework Statement

a) Alice is observing a small ball of mass m in relativistic motion
bouncing elastically back and forth between two parallel walls separated by a distance L
with speed u. After each collision it reverses
direction, thereby creating a clock. What does Alice observe as the
time,$t_A$, between each “tick” of her clock?
(b) Add three events to the space-time diagram: the first when the
ball leaves one side of the box, the second when the ball (first) ar-
rives at the opposite side, and the third when the ball (first) returns
to its initial position. Label the events 1,2 and 3. Connect the events by the world-line
of the ball.
Bob is in an IRF which is moving with speed v in the same direction as the initial motion of the ball. Add this IRF to your space-time diagram. Assume that v<u.
(c) What is the spatial separation, that Bob measures between events 1 and 2?
(d) What is the temporal separation, that Bob measures between events 1 and 2?
(e) What is the spatial separation that Bob measures between events 2 and 3?
(f) What is the temporal separation that Bob measures between events 2 and 3?

## Homework Equations

Lorentz transform

## The Attempt at a Solution

a) This is easily seen to be $t_A =2L/u$.
b) This is a triangle connecting events 1,2 and 3.
c) Using the Lorentz transform Bob would measure: $\Delta x_{12} = \gamma (\Delta x_A -v \Delta t_A)$. Alice measures the spatial separation to be $L$ and the temporal separation to be $L/u$. This seems correct because if I set $v=u$, so that I'm in the rest of the ball on its out-going journey, then the spatial separation is $0$, which makes sense.
d) Using a Lorentz transform: $\Delta t_{12} = \gamma (\Delta t_A -\frac{v}{c^2} \Delta x_A) = \gamma (\frac{L}{u} - \frac{v}{c^2} L)$. Again checking with the ball's rest frame, I get regular time dilation which makes sense.
e) For parts e) and d) I am not sure how to account for the fact the ball's velocity is now $-u$. How do I use this information to find the relevant intervals?

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So how would I implement this as equations? Something like the ball travels from $(L,L/u)$ in Alice's coordinates to $(0,2L/u)$ and then transform these points to Bob's reference frame and use the velocity of the ball as seen by Bob? Or is there something else I'm missing?