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MCQEasyJEE 2026Centre of Mass

JEE Physics 2026 Question with Solution

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 with respect to the origin and the kinetic energy of all the particles with respect to the center of mass as reference.

In the light of the above statements, choose the correct answer from the options given below:

  • A

    Both Statement I and Statement II are true

  • B

    Both Statement I and Statement II are false

  • C

    Statement I is false but Statement II is true

  • D

    Statement I is true but Statement II is false

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Two statements about the total kinetic energy of a system of many particles.

Find: Which statement is true.

Step 1: Analyze Statement I.

For a system consisting of many particles, the total kinetic energy of the system is defined as the sum of the individual kinetic energies of all the particles.

Ktotal=∑12mivi2K_{\text{total}} = \sum \frac{1}{2} m_i v_i^2

Hence, Statement I is true.

Step 2: Analyze Statement II.

The kinetic energy of a system of particles can be split into two parts:

Ktotal=12MVCM2+∑12mivi′2K_{\text{total}} = \frac{1}{2} M V_{\text{CM}}^2 + \sum \frac{1}{2} m_i v_i'^2

where MM is the total mass, VCMV_{\text{CM}} is the velocity of the center of mass with respect to the origin, and vi′v_i' is the velocity of the ii-th particle with respect to the center of mass.

This is a standard result in mechanics. Hence, Statement II is also true.

Conclusion: Both Statement I and Statement II are true.

Therefore, the correct option is A.

Conceptual Explanation

Given: A many-particle mechanical system.

Find: Whether each statement about kinetic energy is correct.

Statement I is true because the definition of total kinetic energy for a system is the sum of the kinetic energies of all constituent particles.

Statement II is also true because total kinetic energy can be decomposed into:

  1. kinetic energy of motion of the center of mass, and
  2. kinetic energy of particles relative to the center of mass.

So the same total kinetic energy may be written in two equivalent forms: as the sum over particle kinetic energies directly, or as translational kinetic energy of the center of mass plus internal kinetic energy relative to the center of mass.

Therefore, both statements are correct, and the correct option is A.

Common mistakes

  • Assuming that the center of mass form of kinetic energy replaces the particle-wise sum. This is wrong because both expressions represent the same total kinetic energy in different forms. Use the decomposition relation as an equivalent rewriting, not a different quantity.

  • Confusing velocity with respect to the origin and velocity with respect to the center of mass. This is wrong because Statement II uses both reference frames carefully. Distinguish VCMV_{\text{CM}} from the relative particle velocities vi′v_i'.

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