How Does Relativity Affect Time and Space Measurements Between Events?

[ZanyCat]

The question...

Two events are observed in a frame of reference S to occur at the same space point, with the second event occurring after a time of 1.70s. In a second frame S' moving relative to S, the second event is observed to occur after a time of 2.25 s.
What is the difference Δx between the positions of the two events as measured in S'?

I know that you guys like people to post their attempt at a solution, but I am legitimately lost and don't even know where to start :(
All I've managed to work out is that for S', event 2 occurred $$6.75*10^8$$m away.

Help would be much appreciated.

 

[voko]

You could start by writing down the relevant equations.

 

[ZanyCat]

That's the problem, I don't know which equations I need. Do I need time dilation? Do I need to consider length contraction? Do I need to use a Lorentz transformation on the coordinates of something? Do I need to get a Lorentz transformation of a velocity?

I am completely lost with this question in every sense of the word.

 

[voko]

The problem is obviously talking about time between two events in one frame and in another frame. What equation is relevant there? What can you determine from it?

 

[Nickie]

I understand your struggle and am happy to provide some guidance. First, let's define some key concepts.

Spacetime is a mathematical concept that combines the three dimensions of space and the one dimension of time into a four-dimensional continuum. In this context, an event is a specific point in both space and time.

In the scenario described, there are two frames of reference, S and S', which are moving relative to each other. This means that observations made in one frame may appear different in the other due to the relative motion between them.

Based on the information given, we can use the formula for time dilation to calculate the difference in time between the two frames. Time dilation states that the time interval between two events is longer in a moving frame than in a stationary frame. This can be expressed as:

Δt' = γΔt

Where Δt' is the time interval in the moving frame, Δt is the time interval in the stationary frame, and γ is the Lorentz factor, which is equal to 1/√(1-(v/c)^2), where v is the relative velocity between the frames and c is the speed of light.

Using this formula, we can calculate the Lorentz factor for the given scenario:

γ = 1/√(1-(v/c)^2) = 1/√(1-(2.25/3)^2) = 1.33

Now, we can use the calculated Lorentz factor to find the difference in time between the two frames:

Δt' = 1.33 * 1.70s = 2.26s

This means that in frame S', the time interval between the two events is 2.26s, which is slightly longer than the 1.70s observed in frame S.

To find the difference in position between the two events, we can use the formula for length contraction, which states that the length of an object in a moving frame is shorter than in a stationary frame. This can be expressed as:

Δx' = (1/γ)Δx

Where Δx' is the length in the moving frame, Δx is the length in the stationary frame, and γ is the Lorentz factor.

Using the Lorentz factor calculated earlier, we can find the difference in position between the two events: