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.

#### Options

A implies B and B implies A.

A does not imply B and B does not imply A.

A implies B but B does not imply A.

B implies A but A does not imply B.

#### Solution

B implies A but A does not imply B.

If the linear momentum of a system is zero,

\[\Rightarrow m_1 \vec{v}_1 + m_2 \vec{v}_2 + . . .\] =0

Thus, for a system of comprising two particles of same masses,

\[\vec{v}_1 = - \vec{v}_2\] ...(1)

The kinetic energy of the system is given by,

\[K . E . = \frac{1}{2}m \vec{v}_1^2 + \frac{1}{2}m \vec{v}_2^2\]

Using equation (1) to solve above equation, we can say:

\[K . E . \neq 0\]

i.e A does not imply B .

Now,

If the kinetic energy of the system is zero,

\[\Rightarrow \frac{1}{2}m \vec{v}_1^2 + \frac{1}{2}m \vec{v}_2^2 = 0\]

\[v_1 = \pm v_2\]

On calculating the linear momentum of the system, we get:

\[\vec{P} = m \vec{v}_1 + m \vec{v}_2 \]

\[\text{ taking v_1 = - v_2 , we can write:} \]

\[ \vec{P} = 0\]

Hence, we can say, B implies A but A does not imply B.