Show that Eq. (26-11) reduces to the nonrelativistic relationship between momentum and kinetic e

Show that Eq. (26-11) reduces to the nonrelativistic...

Key Concepts

Relativistic Energy-Momentum Relation

This concept involves the fundamental relationship between a particle's total energy, its momentum, and its rest mass, expressed by the equation E = sqrt((pc)^2 + (m?c²)²). It serves as the starting point from which kinetic energy is derived by subtracting the rest energy from the total energy. This relationship is pivotal in both relativistic and nonrelativistic physics for understanding how energy and momentum are intertwined.

Nonrelativistic Limit and Series Expansion

When dealing with systems where the kinetic energy is much smaller than the rest energy (K ? m?c²), the square-root expression in the energy-momentum relation can be approximated using a binomial or Taylor series expansion. This approximation simplifies the expression to yield the classical kinetic energy formula, K ? p²/(2m), showing how relativistic formulas reduce to their nonrelativistic counterparts in the appropriate limits.

Rest Energy

Rest energy, defined as E? = m?c², is the inherent energy of a particle due to its mass, independent of its motion. In the nonrelativistic limit, the rest energy dominates over the kinetic energy. Recognizing this allows one to expand the total energy around the large rest energy, thereby isolating the relatively small kinetic energy component, which simplifies the analysis and leads to the classical kinetic energy expression.

Transcript

00:01 In this problem, we're looking at this relativeistic expression between momentum, kinetic energy, rest energy, and under the condition that the kinetic energy is much, much less in the rest energy, like one compared to $100 billion, very, very small, very small, not just less than, but much, much less than.

00:23 Now, let's say you set up an experiment.

00:26 Choose your mass, so you have a certain rest energy, you set up so it has certain kinetic energy, then you run your experiment and you get a certain value for your momentum.

00:36 Your equipment gives you a certain value.

00:38 Then you come over here and you punch in your values for k and e0 and you say, well, how is it compared to the value you got experimentally? say, oh, that's great.

00:48 It gave me the exact same value.

00:50 Wonderful.

00:52 Everything went great.

00:54 Then part of the instructions were to say you calculate the momentum using this formula, which is the classical.

01:03 Relationship between the kinetic energy and the magnet of the momentum.

01:09 And you punch in, you punch in your k and your m here.

01:12 Say, wait a second, it gave me the exact same value.

01:16 I thought these were two different theories.

01:18 Why is that happening? so let's look at this formula here now.

01:29 So let's do some manipulations.

01:32 So we got p .c.

01:33 Squared.