Does Zero Linear Momentum Always Mean Zero Kinetic Energy?

[AdityaDev]

"if kinetic energy of system of particles is zero, then linear momentum of that system of particles is zero but the reverse is not true. That is if linear momentum of a system of particles is zero, then the kinetic energy may not be zero"

This is what I got from my text. Can you provide me with some examples because I am a bit confused. Can you provide some explanation?
Is this true for angular momentum?

 

[Vanadium 50]

Consider two particles with mass m, one with velocity +v and one with velocity -v.

 

[AdityaDev]

Consider two particles with mass m, one with velocity +v and one with velocity -v.

So momentum is zero and kinetic E is not zero. Can you give an example for kinetic energy zero and momentum non zero?

 

[Cruikshank]

So momentum is zero and kinetic E is not zero. Can you give an example for kinetic energy zero and momentum non zero?

KE = sum of 1/2 mv². Can any of those terms be negative?
Or, imagine covering the whole system with a curtain or box and treating it as one object of total mass M and velocity V.
P=MV, while KE= 1/2MV²+ KE(within system, relative to the center of mass)
If KE = 0, what does that imply? Assume masses are never negative.

 

[AdityaDev]

I understood. Thanks @Vanadium 50 and @Cruikshank.