Just what is "energy"?

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For now let's stick with kinetic energy at a normal (non-relativistic) scale. Why does 1kg moving at 2m/s have the same energy as 4kg moving at 1m/s? My intuition, however, is that the latter could do twice as much "work" as the former. Is there any physical or mathematical intuition behind energy that could help here? Is any form of energy other than heat a real "thing" that you can visualize, or is it all just a useful abstraction?