Given below are two statements:
Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.
Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference.
In the light of the above statements, choose the correct answer from the options given below :

Consider two statements regarding the kinetic energy of a mechanical system of many particles. Statement I asserts that the total kinetic energy of the system is the sum of the kinetic energies of all the particles. Statement II claims that the total kinetic energy can also be expressed as the sum of the kinetic energy of the centre of mass with respect to the origin and the kinetic energy of all the particles with respect to the centre of mass.

By definition, the total kinetic energy of a system of $$N$$ particles is given by $$K_{\text{total}} = \sum_{i=1}^{N} \frac{1}{2}m_i v_i^2$$. This expression clearly shows that the total kinetic energy is simply the sum of the individual kinetic energies of all particles. Statement I is true.

The decomposition described in Statement II is known as Konig’s theorem (or the decomposition theorem for kinetic energy), which states that $$K_{\text{total}} = \frac{1}{2}Mv_{\text{cm}}^2 + \sum_{i=1}^{N} \frac{1}{2}m_i v_i'^2$$, where $$M$$ is the total mass of the system, $$v_{\text{cm}}$$ is the velocity of the centre of mass, and $$v_i'$$ is the velocity of the $$i$$-th particle relative to the centre of mass. The first term represents the kinetic energy of the centre of mass motion, while the second term corresponds to the kinetic energy of the particles measured in the centre of mass frame. Statement II is true.

The correct answer is Option (3): Both Statement I and Statement II are true.