1000 Solved Problems in Modern Physics

314 6 Special Theory of Relativity

Transformation of velocities
Differentiating (6.1) with respect to time and noting thatt′=tandνis constant

dx′

dt

=

(

dx

dt

)

−ν (6.5)

U′=U−ν

or

V=U′+ν (6.6)

Invariance of Newton’s second law of motion

InS, force is given by

F=ma=

md^2 x

dt^2

=m

d

dt

(

dx

dt

)

=m

du

dt

InS′, force is given by

F′=

md^2 x

dt^2

=

md^2 x

dt^2

=

md^2 x

dt^2

− 0 =F (6.7)

Galilean Relativity fails for Electromagnetism as evidenced by the negative result
of Michelson–Morley experiment to measure earth’s velocity in the hypothetical
medium of ether.
Einstein took the view that the principle of Relative is correct but time is not
absolute but only relative.

Postulates of special Theory of relativity

(1) The laws of physics apply equally well in all inertial frames of reference, that
is no preferred system exists (the principle of relativity)
(2) The speed of light in free space has the same valuec(= 3 × 108 ms−^1 )inall
inertial frames (the principle of constancy of light)

Lorentz Transformations

x′=γ(x−νt) (6.8)

y′=y (6.9)

z′=z (6.10)

t′=γ

(

t−

νx

c^2

)

(6.11)