To get a long, heavy train moving, the locomotive often…

To get a long, heavy train moving, the locomotive often first pushes back, then pulls forward, repeating this if needed. Explain this procedure.

Explanation of Method

Why a locomotive backs up before starting.

Step 1: Slack in the couplings

The cars are joined by couplings with some play, and their buffers are springs. When the locomotive backs up a little, it pushes all the cars together and closes every gap.

Step 2: Starting the cars one at a time

When the locomotive then pulls forward, it first moves only itself. The gap to the first car closes, and that car is jerked into motion; then the next gap closes and the next car starts, and so on. At each moment the locomotive has to set in motion one car, not the whole train.

Step 3: Why this helps

A car at rest has to be pulled free against the static friction (starting resistance) of its wheels and axle bearings, which is larger than the rolling resistance once it moves. Starting the cars one after another means this larger force has to be overcome only for one car at a time.
The part of the train that is already moving carries momentum; when the next coupling tightens, that momentum helps to jerk the next car into motion.
The force the locomotive can exert is limited by the grip of its wheels ($\mu$ times its weight). Pulling the whole train at once could exceed it and make the wheels spin.

In short: the train is started car by car, so the locomotive never has to break the whole train loose at once.

$\text{1. backing up: all couplings pushed together}$ $\text{2. pulling: the cars start one after another}$ $\text{moving}$ $\text{still at rest}$

Backing up closes the slack in every coupling. When the locomotive then pulls, each coupling tightens only after the previous car is already moving, so the locomotive never has to start the whole train at once.

Result

The train is started car by car, so the locomotive never has to break the whole train loose at once.

Remark

About the original solution. The idea is right, but it has the order mixed up (the original says the locomotive "first pushes backward" in one place and "forward" in another) and the reason slightly off: the point is not to spread the force over time, but to overcome the starting resistance of one car at a time, helped by the momentum of the cars already moving.