Answer
Yes, in a completely inelastic collision between 2 objects where the objects stick together after the collision, it is possible for the final kinetic energy to be zero.
If the final kinetic energy is zero, the initial momentum of the system must be zero. The initial kinetic energy is not necessarily equal to zero.
Work Step by Step
Momentum is a vector, so 2 moving objects can have equal and opposite momentum vectors that sum to zero. For example, consider 2 identical objects, one moving east and one moving west at the same speed. Because they are moving, the system momentum is zero. After they collide and stick, they must be at rest (because now the momentum of this single object is zero). The final kinetic energy is zero.
Kinetic energy cannot be negative. If the final kinetic energy is zero, each object must have zero kinetic energy $\frac{1}{2}mv^2$. Both particles are stationary, so the final system momentum is zero. This equals the initial system momentum, so that was also zero. However, having zero final kinetic energy does not necessarily prove that the initial kinetic energy had to be zero. Reason: we just gave an example where 2 moving objects (i.e., each with positive kinetic energy) collided head-on, and had zero final kinetic energy.
