An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a 500 -m-long train simultaneously at the instant the middle of the train passes him at \(50 \mathrm{m} / \mathrm{s}\). Use the Lorentz transformation to find the time between the lightning strikes as measured by a passenger seated in the middle of the train.

In the observer's frame of reference, the two events (lightning strikes) occur at the same time. As given, he sees the middle of the train (length L = 500 m) pass by him when this happens. Since the train is moving at a speed of 50 m/s, we can calculate the position of each event relative to the observer.

According to the observer, let the events A and B occur at positions \(x_A = -\frac{L}{2}\) and \(x_B = \frac{L}{2}\), respectively, both at time \(t = 0\). Here, \(x_A = -250~m\) and \(x_B = 250~m\).

We know the Lorentz transformations are given by: \[x' = \gamma (x - vt)\] \[t' = \gamma (t - \frac{vx}{c^2})\] Here, \(x'\) and \(t'\) correspond to the coordinates of the event in the frame of the passenger sitting in the middle of the train, \(\gamma\) is the Lorentz factor given by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), \(v\) is the relative speed between the two frames, and \(c\) is the speed of light. First, we calculate the Lorentz factor using the given speed of the train, \(v = 50~m/s\): \[\gamma = \frac{1}{\sqrt{1 - \frac{(50~m/s)^2}{(3 \times 10^8~m/s)^2}}}\approx 1\] Now, we can use the transformation equations to find the coordinates of the events A and B in the train frame: Event A: \[x_A' = \gamma (x_A - vt) = -250~m\] \[t_A' = \gamma (t - \frac{vx_A}{c^2}) = 0\] Event B: \[x_B' = \gamma (x_B - vt) = 250~m\] \[t_B' = \gamma (t - \frac{vx_B}{c^2}) \approx 0.00000167~s\]

Now that we have the time coordinates of both events in the train frame of reference, we calculate the time difference as measured by the passenger in the middle of the train: \[\Delta t' = t_B' - t_A' \approx 0.00000167~s\] So, the time between the lightning strikes as measured by a passenger seated in the middle of the train is approximately 0.00000167 seconds.