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I have a problem with mechanism. Assuming momentum conservation then the 1
kg block moves backwards with speed v.
This also conserves the CM motion so we can deduce no external forces.
However, the brief collision of the objects should be viewed as
adiabatic so no thermal energy (or internal energy) could be transferred
{or as Jim would prefer, one object could not have heated the other} but
somehow all of the KE of the 2 kg block must now be internal energy
while there is no change in energy of the 1 kg block.
So to answer your questions using the (pseudo) work energy theorem, no
work was done on the 1 kg block and a negative mv^2 worth was done on
the 2 kg block
(but by what?). But I can't imagine how such a collision could happen.