Centre of Mass and Collision Part 2

Concept:The total kinetic energy of a system of particles can be expressed either as the sum of individual kinetic energies or as the sum of the kinetic energy of the centre of mass and the kinetic energy relative to the centre of mass.Explanation:Statement I is true because the total kinetic energy $K_{\text{total}}$ of a system of $n$ particles is simply the algebraic sum of the kinetic energies of each particle.If particle $i$ has mass $m_i$ and velocity $v_i$ in a given frame, then $K_{\text{total}} = \sum_{i=1}^{n} \frac{1}{2} m_i v_i^2$.Statement II is also true. The total kinetic energy can be split into two parts: 1. The kinetic energy of the centre of mass: $K_{\text{cm}} = \frac{1}{2} M_{\text{total}} V_{\text{cm}}^2$, where $M_{\text{total}}$ is the total mass and $V_{\text{cm}}$ is the velocity of the centre of mass relative to the origin.2. The internal kinetic energy: $K_{\text{internal}} = \sum_{i=1}^{n} \frac{1}{2} m_i v_{iC}^2$, where $v_{iC}$ is the velocity of particle $i$ relative to the centre of mass.Thus, $K_{\text{total}} = K_{\text{cm}} + K_{\text{internal}}$.Hence, both statements are correct.Answer:Both Statement I and Statement II are true – Option D.