Relativity: Find the magnitude and direction

[endusto]

Homework Statement

An observer in frame S standing at the origin observes two flashes of colored light separated spatially by Δx = 2400 m. A blue flash occurs first, followed by a red flash 5 μs later. An observer in S' moving along the x-axis at speed v relative to S also observes the flashes 5 μs apart and with a separation of 2400 m, but the red flash is observed first. Find the magnitude and direction of v.

Homework Equations

v = d/t
c = 3*10^8

The Attempt at a Solution

t2 = time until observer saw blue light
t1 = time until observer saw red light
c*t1 + c*t2 = 2400m = Δx (1)
t1 = t2 + 5μs (2)

substituting t1 with t2 + 5μs into (1):
c*(t2 + 5μs) + c*t2 = 2400m
c*(t2 + t2) = 2400m - 5μs*c
2*t2 = (2400m - 1498.96229m)/c
2*t2 = (901.03771m)/c
t2 = 901.03771m/c/2
t2 = 1.50276914 microseconds
t1 = t2 + 5μs
t1 = 6.50276914 microseconds

so basically all i found was the time it took the red and blue flash to reach the person. i don't know if i was supposed to do that. i don't really know what to do now so help is very much appreciated.

 

[anathema]

First, we need to find the distance the observer in S' is from the origin, which we can call x'. We can use the equation for time dilation to relate the time experienced by the observer in S' (t') to the time experienced by the observer in S (t):

t' = t/√(1-v^2/c^2)

Since both observers see the flashes 5 μs apart, we can set t' = 5 μs and t = 6.50276914 μs (as you found in your attempt at a solution). Plugging these values into the equation, we get:

5 μs = 6.50276914 μs/√(1-v^2/c^2)

Rearranging for v, we get:

v = c√(1 - (5 μs/6.50276914 μs)^2) = 2.586*10^8 m/s

To find the direction of v, we can use the equation for the Lorentz transformation of velocities:

v' = (v - u)/(1 - vu/c^2)

Where u is the velocity of S' relative to S (in this case, u = v). Rearranging for v', we get:

v' = (v + u)/(1 + vu/c^2) = 2.586*10^8 m/s

Since v' is in the opposite direction of v, the direction of v is opposite to the direction of the observer in S'. So, the magnitude of v is 2.586*10^8 m/s and its direction is opposite to that of the observer in S'.

 

[tomcue]

I would first clarify the problem by asking for more information. In order to find the magnitude and direction of v, we need to know the speed of the observer in frame S' and the direction in which they are moving along the x-axis. Without this information, we cannot provide a complete response to the problem.

However, based on the information given, we can make some observations. The fact that the red flash is observed first in frame S' suggests that the observer is moving towards the source of the light. Additionally, the flashes are observed at the same spatial separation and time interval in both frames, indicating that the observer in S' is moving at a constant velocity relative to S.

To find the magnitude and direction of v, we would need to use the Lorentz transformation equations, which describe how measurements of space and time differ between two frames of reference moving at a constant velocity relative to each other. These equations involve the speed of light (c) and the Lorentz factor (γ), which is dependent on the relative velocity (v) and can be calculated using the equation γ = 1/√(1-(v^2/c^2)). With this information, we can solve for the magnitude and direction of v.

In summary, in order to fully answer this problem, we would need to know the speed and direction of the observer in frame S'. Without this information, we can make some assumptions and observations, but we cannot provide a complete response.