If a system of particles has zero momentum, is the kinetic energy of the system necessarily zero

step 1 here for the solution step one the momentum is a vector quantity so the momentum may have value

If a system of particles has zero momentum, is the kinetic...

Key Concepts

Center-of-Mass Frame

The center-of-mass frame of a system is defined as the frame where the total momentum is zero. However, particles within the system can still be in motion relative to the center of mass, contributing to the system's overall kinetic energy. Conversely, if all particles are at rest relative to an observer, both the system's kinetic energy and momentum are zero.

Vector versus Scalar Quantities

The distinction between vector and scalar quantities is crucial. Momentum, being a vector, has both magnitude and direction, allowing for cancellation when summed. Kinetic energy, on the other hand, is a scalar, meaning it only has magnitude and all contributions add up regardless of direction. This difference explains why a zero net momentum does not necessarily imply zero kinetic energy.

Momentum

Momentum is a vector quantity representing the quantity of motion of a particle or system of particles. For a system, it is the vector sum of the individual momenta, which means that even if the total (net) momentum is zero, individual particles may still be moving in different directions with nonzero momenta that cancel each other out.

Kinetic Energy

Kinetic energy is a scalar quantity that measures the energy of motion of a particle or system of particles. It is calculated as the sum of the kinetic energies of the individual particles, and it is always a positive quantity (or zero in the case of rest). Unlike momentum, it does not cancel out through vector addition.

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