What is the motion of K' in the direction of the x-axis?

[xWaffle]

Homework Statement

Find a system K' in which the following events in frame K appear at the same time:

i) x₁ = a; t₁ = (2a) / c; y₁ = 0; z₁ = 0
ii) x₂ = 2a; t₂ = (3a) / (2c); y₂ = 0; z₂ = 0

Describe the motion of K'.

Homework Equations

Lorentz transformations; time dilation; length contraction..maybe?

The Attempt at a Solution

My scratch-work is too complicated to try and re-post here, but basically I have come to the conclusion I don't really know how to use the equations to describe the motion of K'. I need to describe the motion of K' in the direction along which the motion occurs, which is obviously the x-axis

 

[voko]

What is the time of any event in K as seen in K'?

 

[xWaffle]

In K the event is seen as T0 = t2 - t1 = (3a / 2c) - (2a / c), and in K'.. um, T' = gamma*T0? If that's so then I don't know how to calculate this because gamma relies on velocity which is what I'm trying to find

 

[voko]

No, it is not gamma*T0 in K'. What is the Lorentz transformation of the temporal coordinate?

 

[Nickie]

.

Based on the given events in frame K, we can use the Lorentz transformations to find the coordinates of the events in frame K'. The Lorentz transformations describe how coordinates in one frame of reference (K) are related to coordinates in another frame (K') that is moving at a constant velocity relative to K.

In this case, we can use the time dilation equation to find the time coordinate (t') in frame K' for both events:

t' = γ(t - vx/c^2)

Where γ is the Lorentz factor, v is the velocity of frame K' relative to frame K, t is the time coordinate in frame K, and c is the speed of light.

Using the given values, we can plug in the values for t1 and t2 to find the corresponding t' values for both events. We can then use the length contraction equation to find the x' coordinate in frame K' for both events:

x' = γ(x - vt)

Where x is the x-coordinate in frame K, and t is the time coordinate in frame K.

Again, we can plug in the given values for x1 and x2 to find the corresponding x' values for both events. This will give us the coordinates of the events in frame K'.

The motion of K' in the direction of the x-axis can then be described as moving at a constant velocity (v) relative to frame K, with the events occurring at the same time in frame K'. The specific value of v can be calculated using the Lorentz transformations and the given coordinates.

In summary, the motion of K' can be described as moving at a constant velocity relative to frame K, with the events occurring at the same time in frame K'. This is in accordance with the principle of relativity, which states that the laws of physics should be the same in all inertial frames of reference.