accelerating a train

There is a nice old problem of the force exerted by an engine pulling a set of train cars, all of which are initially at rest on a frictionless track, and all of which end up moving at speed v. You can think about it in terms of momentum (the change in momentum is due to a force acting over a time) or in terms of kinetic energy (the kinetic energy is produced by a force acting over a distance). The two ways of thinking about it get different answers by a factor of two!

The resolution is simple: The momentum of the train must be created by a force, and since it is a vector law in a one-dimensional problem, the total force times the total time must equal the total momentum. Any other solution fails to conserve momentum. The work done by the engine can go into kinetic energy, but it can also go into other forms of energy (like oscillation or dissipation in the train linkages). Energy is conserved as long as the kinetic energy is less than the work done. This resolves the discrepancy, and creates the nice result that when a train is accelerated, there must be dissipation, or else the train will be left vibrating or oscillating as it goes. I think this is an example of impedance matching.