An electron has a speed of 0.750 c . (a) Find the speed of a...

An electron has a speed of $0.750 c .$ (a) Find the speed of a proton that has the same kinetic energy as the electron. (b) What If? Find the speed of a proton that has the same momentum as the electron.

Key Concepts

Relativistic Kinetic Energy

In special relativity, kinetic energy is defined using the Lorentz factor as K = (? - 1)mc², where ? = 1/?(1 - v²/c²). This expression accounts for the increased energy required to accelerate an object as its speed approaches the speed of light, differing from the classical ½mv² form, and is crucial when dealing with particles moving at significant fractions of the speed of light.

Relativistic Momentum

Relativistic momentum is given by p = ?mv, incorporating the Lorentz factor to adjust for the effects of high velocities. Unlike classical momentum, this formulation ensures that momentum increases without bound as an object’s speed nears the speed of light, reflecting the fact that no massive particle can reach or exceed the speed of light.

Lorentz Factor

The Lorentz factor, denoted as ?, is a critical component in special relativity defined as ? = 1/?(1 - v²/c²). It quantifies how much time, length, and relativistic mass differ for observers in different inertial frames and is key to modifying classical formulas for energy and momentum in high-speed regimes.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated in the equation E = mc², highlights the intrinsic relationship between mass and energy. This principle underpins the modifications to kinetic energy calculations at relativistic speeds, illustrating how a particle’s rest mass contributes to its total energy, and is fundamental when comparing the energy properties of particles with different masses.

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