Topic: Energy
Concepts: 1D elastic collision, 1D inelastic collision, Energy dissipation

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Solution:

For the perfectly inelastic collision, we conserve linear momentum to find the velocity of both objects after the collision,

Since the combined mass breaks the cord and ends up at rest, all the kinetic energy became spring potential energy (right before it breaks),

In other words, it costs energy $U$ to break the spring. For the elastic collision, recall the elastic collision equations for a mass $m$ colliding with $M$ at rest,

In our case, we have

Since the mass $3m$ breaks the cord and ends up at speed $v_f$, we conserve energy to obtain

Substituting our earlier equations,

so the answer is C.

Topic: Energy
Concepts: Conservation of energy, Energy dissipation

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Solution:

The initial kinetic energy of the system is

The final kinetic energy after the elastic collision and the cord breaking is

but since an elastic collision conserves energy, the final kinetic energy is also equal to

where $U$ is the energy cost to break the spring. From the previous problem, we found

Thus,

so the answer is D.