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प्रश्न
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.
पर्याय
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.
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उत्तर
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.
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