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.