Question

Statement-1: Kinetic energy of system = \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2 \) Statement-2: Kinetic energy of system = Kinetic energy of center of mass + kinetic energy with respect to center of mass.

• A. Statement I is true; Statement II is true
• B. Statement I is true; Statement II is false
• C. Statement I is false; Statement II is true
• D. Statement I is false; Statement II is false

Hint:

The total kinetic energy of a system is the sum of the kinetic energy of the center of mass and the kinetic energy relative to the center of mass.

Answer 1

The Correct Option is A

Step 1: Analyze Statement 1. 
The total kinetic energy of a system of particles is the sum of the kinetic energies of each particle. This matches Statement 1: \[ KE_{\text{total}} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2. \] Thus, Statement 1 is true. 
Step 2: Analyze Statement 2. 
The kinetic energy of a system can also be described as the kinetic energy of the center of mass (which is \( \frac{1}{2} M V_{\text{cm}}^2 \)) plus the kinetic energy due to the motion of particles relative to the center of mass. This matches Statement 2. 
Step 3: Conclusion. 
Both Statement 1 and Statement 2 are correct. 
Final Answer: \[ \boxed{\text{Statement I is true; Statement II is true.}} \]