Relativity, difference in x given t

[ProPatto16]

Homework Statement

two events are observed in frame of reference S to occur at the same point. the second event is 1.8s after the first. in a second frame S' moving relative to S, the second event is 2.35 s after the first. what is the difference between the positions of the two events as measured in S?

Homework Equations

t'=(t-ux/c²) / sqrt(1 - u²/x²)

The Attempt at a Solution

t = 1.8, t' = 2.35
need to solve for x but the variable u is also unknown?

 

[ProPatto16]

also set it up so S=0=S' and t=0=t' meaning S and S' begin in the same place at the same time, where and when event 1 occurs.

 

[Doc Al]

You wrote the equation for t'. Write the corresponding one for x'. That will allow you to solve for gamma and u.

Even easier would be to compare the invariant spacetime interval between the events.

 

[ProPatto16]

what do you mean by invariant spacetime interval? is that the difference between 1.8 and 2.35?

and using the x' formula, solving for u then subbing into t' then gives you the unknown x' so how does that work? cause there's no way to find a numerical answer for u so i need to cancel it out or sub right?

 

[Doc Al]

what do you mean by invariant spacetime interval? is that the difference between 1.8 and 2.35?

No, the spacetime interval is given by:

$$(\Delta s)^2 = (\Delta x)^2 - c^2 (\Delta t)^2 = (\Delta x')^2 - c^2 (\Delta t')^2$$

See: http://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals"

and using the x' formula, solving for u then subbing into t' then gives you the unknown x' so how does that work? cause there's no way to find a numerical answer for u so i need to cancel it out or sub right?

Using the equation you have for t', you can solve for gamma (and the speed if you like). You'll need the corresponding equation for x' in terms of t and x to find x'.

 

[ProPatto16]

i must still be missing something. using the formulas:


t'=(t-ux/c²) / sqrt(1 - u²/x²)

and

x'=(x-ut) / sqrt(1 - u²/x²)

the only known quantities are t and t'
that leaves the unknowns to be x, x' and u.

conditions are

at t=0 x=x' and u=0 since S=S'

3 unknowns and 2 formulas. can't solve for them.

im so lost.
forget the spacetime interval way, ill do it the formula way cause its what I've been doing, but i just always seem to end up with 2 unkowns left no matter how i reaarange?

 

[Doc Al]

the only known quantities are t and t'
that leaves the unknowns to be x, x' and u.

There's what you're missing. You are given x:

two events are observed in frame of reference S to occur at the same point.

So what's x?

 

[ProPatto16]

zero. so blind.

so 2.35 = 1.8 / sqrt(1-u²/c²)

so u=0.64289c

sub into eqn for x'

x' = 0 - 0.64289c*1.8 / sqrt(1-0.64289²c/c)

= -452935555.9

yes? that would be in metres?

 

[ProPatto16]

that can't be right. that's more distance than light travels in 1 second.

 

[Doc Al]

that can't be right. that's more distance than light travels in 1 second.

What's that got to do with anything?

(Your answer looks fine. Try using the invariant interval method as a check.)

 

[ProPatto16]

the difference between 1.8s and 2.35s is less then 1 second? how can reference frame S' move that far in less than a second?

 

[Doc Al]

the difference between 1.8s and 2.35s is less then 1 second?

Those times are from different frames. You can't subtract them and get something meaningful.

how can reference frame S' move that far in less than a second?

It doesn't. According to frame S', frame S moves that distance in 2.35s.

 

[ProPatto16]

ohh okay. thanks so much :)
conceptual challenges are not my strongest point.