1000 Solved Problems in Modern Physics

6.1 Basic Concepts and Formulae 317

Rule: Every rigid body appears to be longest when at rest relative to the observer.
When it is moving relative to the observer it appears contracted in the direction of
its relative motion by the factor

√

1 −ν^2 /c^2 , while its dimensions perpendicular to
the direction of motion are unaffected.

Addition of velocities

β=

β 1 +β 2

1 +β 1 β 2

(6.29)

Mass, energy and momentum

m=m 0 γ=

m 0

√

1 −ν^2 /c^2

(6.30)

wheremis the effective mass andm 0 is the rest mass.
The rest mass energy

E 0 =m 0 c^2 (6.31)

The total energy of a free particle is

E=T+m 0 c^2 =mc^2 =m 0 γc^2 (6.32)

whereTis the kinetic energy.
The momentumpis given by

P=mν=m 0 γβc (6.33)

E^2 =c^2 p^2 +m^20 c^4 (6.34)

cp=βE (6.35)

c^2 p^2 =T^2 + 2 Tm 0 c^2 (6.36)

Lorentz transformations of momentum and energy

cpx′=γ(cpx−βE) (6.37)

cpy′=cpy (6.38)

cpz=cpz (6.39)

E′=γ(E−βcpx) (6.40)