Consider the following two statements:

(A) Linear momentum of a system of particles is zero.

(B) Kinetic energy of a system of particles is zero.

Suppose there are n particles in an isolated system. Then,

Now, in real vector space, Mass cannot be negative, and square of velocities can’t be negative too.

So, Kinetic energy zero means, all the masses and velocities are zero. Hence Linear momentum

must be zero too. So, B implies A (proved). Now, Linear momentum zero means, the velocities

may be in opposite direction or masses may change, so no such assumption can be taken. Hence,

A doesn’t imply B.