The train has been modified so that it can only accelerate or de-accelerate.  The train cannot travel at a constant velocity.

The engineer would like the train to have a constant acceleration for the first half, and a constant negative acceleration for the last half.

Ignoring friction or wind resistance, what should he use for a rate of acceleration?  How much energy will the train consume over the entire trip?  If, instead, the train could travel at a constant speed, would the engineer use less fuel if he were to travel the majority of the trip at a constant velocity?  If we account for wind or friction, which train would be more efficient?