Consider the following two statements:

(A) Linear momentum of the system remains constant.

(B) Centre of mass of the system remains at rest.

#### 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.

The centre of mass of a system is given by,

\[\vec{R} = \frac{1}{M} \sum_{} m_i \vec{r}_i\]

On differentiating the above equation with respect to time, we get:

\[\frac{d \vec{R}}{dt} = \frac{1}{M} \sum_{} m_i \frac{d \vec{r}_i}{d t}\]

As the centre of mass of the system remains at rest, we have:

\[\frac{1}{M} \sum_{} m_i \frac{d \vec{r}_i}{d t} = 0\]

\[ \sum_{} m_i \vec{v}_i = 0\]

This implies that the linear momentum of the system remains constant.