Can someone please explain why the distance that the train moves, vt, is not Lorentz Contracted too? How can one section of the train be contracted and not the whole thing?

It is not space that is physically contracting. The Lorentz contraction formula defines the expansion of the metric used to measure space (distance). A metre is defined as the physical distance light travels per second. If I am on the train the seconds on the platform are longer (because the platform is moving with respect to me) so the metre used by someone on the platform is longer and so they will measure the distance to be shorter (that is fewer metres) than what I would measure being on the train.

It is not space that is physically contracting. The Lorentz contraction formula really defines the expansion of the metric used to measure space (distance). A metre is defined as the physical distance light travels per second. If I am on the train the seconds on the platform are longer (because the platform is moving with respect to me) so the metre used by someone on the platform is longer and so they will measure the distance to be shorter (that is fewer metres) than what I would measure being on the train.

It is not space that is physically contracting. The Lorentz contraction formula really defines the expansion of the metric used to measure space (distance). A metre is defined as the physical distance light travels per second. If I am on the train the seconds on the platform are longer (because the platform is moving with respect to me) so the metre length used by someone on the platform is longer and so they will measure the distance to be shorter (that is fewer metres) than what I would measure being on the train.

Hey Jack, it is the train yhat moves, not the distance.

The object in motion is the reason for the season, if it is ok i say that. Lorentz allows for calculations on moving things.

The distance… Well… Is said to not be moving (ignoring planetary, solar system, stellar, Milky Way, local group, local cluster and super virgo cluster speeds. Can those be ignored?)

Lorentz transformations, set of equations in relativity physics that relate the space and time coordinates of two systems moving at a constant velocity relative to each other.