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Why S=x2-x1

Why they have the same time

Hi joseotilio25, you're keeping me busy today. :)
First, your question "Why $S=x_2 - x_1$": $S$ is the separation between the trains, $x_2$ is the position of train 2, and $x_1$ is the position of train 1. The distance between the trains is the difference between their positions. Make sure you're not thinking of the "x's" as representing a distance. Only $S$ is a distance, whereas each $x$ is a position. When you subtract $x_1$ from $x_2$, you get the distance between these positions. Just to illustrate, if one object is at a position of $x_2 = 3 \textrm{ m}$, and a second object is at a position of $x_1 = -4 \textrm{ m}$, their separation would be $3 \textrm {m} - (-4 \textrm{ m}) = 7 \textrm{ m}$.

Your second question "Why they have the same time": We begin looking at both trains at the same time (call it $t_i =0 \textrm{ s}$ if you like), and we need to figure out at what future time they will meet. Since $t_i$ is the same for both trains, and $t_f$ is also the same since they will both report meeting at the same moment (this isn't chapter 27 on relativity, after all), the elapsed time, which is what $t$ represents (perhaps better written as $\Delta t$ which is $t_f - t_i$) is the same for both trains.

Why does X2 = X2i + V2 (T)?

Hi swedesanddanes, I'm guessing that the confusing part is the $x_{2i}$ in $x_2 = x_{2i} + v_2 t$, correct? For most questions it's typically enough to say something like $x_2 = v_2 t$, but the reason this is usually OK is that normally you choose the initial position to be zero, in which case $x_2 = x_{2i} + v_2 t$ would be the same as $x_2 = v_2 t$ since $x_{2i} = 0$ when the initial position is zero. This question, however, is not one of your usual questions. The second train begins at an initial position of $8.5 \textrm{ m}$, in which case $x_{2i} = 8.5 \textrm{ m}$, so this initial position term is needed in the position formula.

Just to check your understanding, notice how we could have instead chosen $x_{2i} = 0$ after all, but then the initial position of train 1 would need to be $x_{1i} = -8.5 \textrm{ m}$ (assuming positive is directed toward the right).

Hope this helps,
Mr. Dychko