A free particle of mass has momentum of magnitude

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime · Relativistic dynamics and fields in flat spacetime

Relativity and Gravitation, Ch Relativistic momentum, energy and collisions

Problem

A free particle of mass

m

has momentum of magnitude

p

. Its kinetic energy is

K = E - mc^2

. ((a)) Using the energy-momentum relation, show that

K = mc^2\left[\sqrt{1 + (p/mc)^2} - 1\right]

. ((b)) Expand this with the binomial series for

p \ll mc

, keeping two terms, and show that

K \approx \dfrac{p^2}{2m} - \dfrac{p^4}{8m^3c^2}

, so that the Newtonian formula

p^2/2m

is too large by the fraction

p^2/4m^2c^2

. ((c)) For an electron (

m_ec^2 = 0.5110\ \mathrm{MeV}

) with

pc = 0.0500\ \mathrm{MeV}

, compute

K

from part (a) and from the Newtonian formula

p^2/2m

, both in keV, and give the percentage by which the Newtonian value is too large. ((d)) Repeat the two calculations of

K

for an electron with

pc = 1.000\ \mathrm{MeV}

, giving both in MeV, and comment.

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