Question

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

Answer 1

No. As the potential energy can have a negative value, the total energy of the system may sum up to zero.

For example:
Two masses A and B having masses 2 kg and 4 kg respectively move with a velocity of 4 ms^-1 in opposite directions.

Kinetic energy of system (A and B)
$$\begin{gathered} =\frac{1}{2}\times 2\times 4^2+\frac{1}{2}\times 4\times 4^2 \\ =48\mathrm{J} \end{gathered}$$

If the gravitational potential energy of the system is −48 J, the total energy of the system will be zero. However, the linear momentum will be non-zero.