Show that the kinetic energy of a particle of mass m is...

Show that the kinetic energy of a particle of mass m is related to the magnitude of the momentum p of that particle by $K E=p^{2} / 2 m .$ (Note: This expression is invalid for particles traveling at speeds near that of light.)

Key Concepts

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. In the classical mechanics framework, it is given by the equation KE = 1/2 mv², where m represents the mass of the object and v its velocity. This concept is fundamental for understanding how energy is transferred in moving systems.

Momentum

Momentum is a measure of the motion of a particle and is defined classically as p = mv, where m is the mass of the object and v is its velocity. It represents the quantity of motion and plays a central role in both conservation laws and dynamics in mechanics.

Relationship Between Kinetic Energy and Momentum

The derivation of the relationship between kinetic energy and momentum involves substituting the expression for velocity from the momentum equation into the kinetic energy equation. By expressing velocity as p/m and substituting it into the kinetic energy formula, you obtain KE = p²/(2m), which shows the direct relationship between these two important physical quantities in non-relativistic mechanics.

Non-Relativistic Approximation

The relationship KE = p²/(2m) is derived under the assumption that the speeds involved are much less than the speed of light. This non-relativistic approximation simplifies the equations of motion and is valid in most everyday phenomena, but it breaks down when particles move at speeds approaching that of light, where relativistic effects become significant.

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