Special relativity, simultaneity

[fluidistic]

Homework Statement

A train of 0.8 km (measured by an observer over the train) travels at a speed of 100 km/h. Two lightnings strike simultaneously the back and the front of the train, according to an observer on the ground. What is the time separating both strikes according to the observer on the train?

Homework Equations

Lorentz transformations.

The Attempt at a Solution

Let O be the reference frame of the observer on the ground and O' be the frame of reference of the observer on the train.
They give me the information $$x_B'-x_A'=800m$$ and $$t_B-t_A=0$$. They ask me $$t_B'-t_A'$$.
Using Lorentz transformations, I get that $$t_B'+\frac{vx_B'}{c^2}-t_A'-\frac{vx_A'}{c^2}=0$$.
I converted the km/h to m/s and my result is that $$t_B'-t_A'=-2.4691358 \times 10 ^{-13}s$$. It means therefore that $$t_A'>t_B'$$, thus the observer on the train sees first the strike of the lightning on the front of the train and then the one on the back of the train and their time separation is about $$2.4691358 \times 10 ^{-13}s$$.
I don't know if it's true. To me it seems a too little time, although I realize that the train is only moving at $$\frac{1000m}{36s}\approx \frac{30m}{s}$$ which is very small compared to c. The order of the strikes seems logical to me...
Can someone confirm/infirm my result?

 

[Doc Al]

Your answer is correct.

 

[fluidistic]

Your answer is correct.

Ok thank you, good to know. Problem solved.