Question

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. 

Answer 1

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.