If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?

Yes, it is possible for the particle to have a non-zero momentum.

Explanation:

Consider the above scenario. The particle is ‘h’ height below the

Earth's surface.

Potential energy, .

Kinetic energy, .

Total energy, .

Let us suppose for some value of height h and velocity v, the total

energy becomes zero. In that case, we have zero total mechanical

energy but non-zero kinetic energy.

Now, non-zero kinetic energy implies that the particle has non-zero

momentum.