(Relativity) Space Time Interval

[ak345]

Homework Statement

Frames S and S' are moving relative to each other along the x and x' axes. They set their clocks to
t = t'=0 when their origins coincide. In frame S, Event A occurs at xA = 1 yr and tA = 1 yr, while event B occurs at xB = 2 yr and tB = 0.5 yr. These events occur simultaneously in S'.
(a) Find the magnitude and direction of the velocity of S' relative to S.
(b) Draw a spacetime diagram to confirm part (a).
(c) At what time do both events occur as measured in S'?
(d) At what locations do the events occur as measured in S'?

Homework Equations

I believe V=x/t can be used
Also, slope=1/v

The Attempt at a Solution

Using those two equations, the velocity for A is 1 and the velocity for B is 4
I don't know how those relate to S and S'. Also, isn't the slope of C=1? So B is going faster than the speed of light? Thats doesn't seem right

 

[BvU]

A and B are events in space-time. There is no movement of A and B. Only S and S' are moving.
Your relevant equations should not be slope (of what?) = 1/v and V = x/t (that looks as if you think A was in the origins when S and S' set their clocks and has moved to x in frame S in one year of S time). I shudder to think what your idea about the velocity of B could possibly be.

X_A=1 yr is also very strange. Do you mean x_A/c = 1 yr ?

You treat C as another event, but the speed of light, c is what it says it is: a speed.

Your relevant equations should have something to do with Lorentz transformations. And I suppose you know about them...
If not, you have to read up, because some relativity is needed for this exercise...

Oh, and: Hello Ak, and welcome to PF !

 

[vela]

Homework Statement

Frames S and S' are moving relative to each other along the x and x' axes. They set their clocks to
t = t'=0 when their origins coincide. In frame S, Event A occurs at xA/c = 1 yr and tA = 1 yr, while event B occurs at xB/c = 2 yr and tB = 0.5 yr. These events occur simultaneously in S'.
(a) Find the magnitude and direction of the velocity of S' relative to S.
(b) Draw a spacetime diagram to confirm part (a).
(c) At what time do both events occur as measured in S'?
(d) At what locations do the events occur as measured in S'?

Homework Equations

$gamma=1/(sqrt(1-beta^2))$

The Attempt at a Solution

I am having a really tough time figuring out what this looks like. These events start in S and end up in S'. S is our fixed frame and S' is our moving frame. How do I calculate the velocity of S'? Do both events end up in S'

The events exist independently of S or S'. When you say $x_A/c = 1\text{ yr}$ and $t_A = 1\text{ yr}$, you're choosing to orient your coordinate system such that those statements are true in frame S.

The Lorentz transformations allow you to calculate the coordinates of an event in a different frame if you know the relative velocity of the two reference frames. This is analogous to finding the coordinates of a point in a rotated system. By this I mean, suppose there's a point that has coordinates (x,y) in your original system, and now you rotate the axes by an angle $\theta$ and you want to find the coordinates of the point (x',y') with respect to the new axes. As you hopefully learned in the past, you simply calculate
\begin{eqnarray*}
x' = x \cos\theta - y\sin\theta \\
y' = x \sin\theta + y\cos\theta
\end{eqnarray*} These equations simply relate the old coordinates (x,y) of a point to the new ones (x',y') given the angle $\theta$. The Lorentz transformations similarly relate space-time coordinates (t,x) of an event in S to the space-time coordinates (t',x') of the event in S'.

It might help you to think about part (b) first. Have you drawn a spacetime diagram for the events in S? What do the x' and t' axes look like on that diagram?