What is the Time Interval Between Arrival of Wave Fronts at C' in S?

[BigFlorida]

Homework Statement

Assume that the train shown in the figure below is 1.9 km long as measured by the observer at C' and is moving at 170 km/h. (The point C' is midway along this train.) What time interval between the arrival of the wave fronts at C' is measured by the observer at C in S?

Homework Equations

c = 3x10^5 km/s[/B]

The Attempt at a Solution

I have been staring at this problem for a few hours and nothing seems to be jumping out at me. I did not know if I was supposed to measure the train with a ruler to get the proper scale (7.2 cm = 1.9 km) and then measure the interval from A' to B' to calculate the actual length (~1.4 km), or to consider the length from A' to B' to be 1.9 km. The former seems like a lot more work than my teacher would expect, but the latter seems much more logical because in the problem statement it says "the length of the train is 1.9 km" and the interval A' to B' is obviously not the length of the entire train.

I know just by thinking about it that the interval should be tiny, but I cannot think about how to calculate it. I think I am missing something glaringly obvious, but I just cannot see it.

To me, it seems a choice for the upper bound, t₂, would be 4.67x10^-6 s because one light flash starts at B' = B and by the time it reaches C', C' = A which implies (to me) that the beam of light has traveled the length of the interval (1.4 km). I could be wrong though.

I still have no way of getting t₁ though.

Also, I do not know if it is proper to believe that C' is halfway in between A' and B' if A' -> B' = 1.4 km, because the front part of the train (to the right of A') is longer than the back side of the train (to the left of B') and C' is the center of the train. So that also greatly throws things off for me..

Thank you in advance for any answers.

 

[Chestermiller]

Where did this problem come from? The problem statement is a disaster on many levels.

Chet

 

[andrewkirk]

My guess is that you are expected to assume (1) C is the mid-point of AB and (2) AB=1.9km. To do otherwise would require measuring the diagram and that is very rarely done in such problems. The problem is poorly stated. They should have explicitly stated (1) and (2), but let's assume they intended to but just forgot.

I don't think the problem actually requires use of any special relativity techniques such as Lorentz contraction or time dilation. The whole problem can be done using classical techniques in the frame S, because it doesn't ask about any distances or times as measured in the train's frame.

To solve it, first let's label a couple more points to make reference easy. Let C1 be the location of C' when the light wave from A reaches C', and let C2 be the location of C' when the light wave from B reaches C'.

If you can work out the distance in S between C1 and C2 then the time interval will be the time C' takes to get from C1 to C2.

To work out the location of C1, make its distance from C a variable x then write an equation based on the fact that C' travels from C to C1 and the light travels from A to C1 in that time. Solve for x.

Then do the same thing for C2.