# Chapter 1:The special theory of relativity¶

## Example no:3,Page no:32¶

In :
#variable declaration
x=50.0
y=20.0
z=10.0                            #x,y,z cordinates in meters(frame s)
t=5.0*10**(-8)                              #time in seconds(frame s)
velocity=0.6*3*10**8                      #velocity of observer in s' frame relative to s in meter/second
c=3.0*10.0**8                                 #speed of light in meter/second
Beta=0.6
Gamma=1.0/((1.0-Beta**2)**(1.0/2.0))

#Calculation
xdash=Gamma*(x-(velocity*t))              #value of x cordinate in frame s' in meters
ydash=y                                   #value of y cordinate in frame s' in meters
zdash=z                                   #value of z cordinate in frame s' in meters
tdash=Gamma*(t-((velocity*x)/(c**2)))      #value of t in frame s' in seconds

#Result
print"\nValue of space time cordinates in frame s:\n\t x=",xdash," m\n\t y=",ydash,"m\n\t z=",zdash,"m\n\t t=",tdash,"s"

Value of space time cordinates in frame s:
x= 51.25  m
y= 20.0 m
z= 10.0 m
t= -6.25e-08 s


## Example no:4,Page no:32¶

In :
#variable declaration
x1=20.0                                         #position of event 1 in meters(frame s)
t1=2.0*10**(-8)                                 #time of event 1 in seconds(frame s)
x2=60.0                                        #position of event 2 in meters(frame s)
t2=3.0*10**(-8)                                 #time of event 2 in seconds(frame s)
c=3.0*10**8                                    #speed of light in meter/second
v=0.6*c                                     #speed of frame s' relative to frame s (meter/second)
Beta=0.6
Gamma=1.0/((1.0-Beta**2.0)**(1.0/2.0))

#Calculation
#part(i)
separation=Gamma*((x2-x1)-v*(t2-t1))        #spatial separation of the events in frame s' (meter)
#part(ii)
interval=Gamma*((t2-t1)-(v*(x2-x1))/(c**2))  #time interval between the two events in frame s' (second)

#Result
print"\nIn frame s:\n\t (i)spatial separation=",separation,"m\n\t (ii)time interval=",interval,"s"

In frame s:
(i)spatial separation= 47.75 m
(ii)time interval= -8.75e-08 s


## Example no:5,Page no:33¶

In :
#variable declaration
x1=24.0                                         #position of event 1 in meters(frame s)
t1=8.0*10**(-8)                                  #time of event 1 in seconds(frame s)
x2=48.0                                         #position of event 2 in meters(frame s)
t2=4.0*10**(-8)                                  #time of event 2 in seconds(frame s)
c=3.0*10**8                                      #speed of light in meter/second

#calculation
v=((c**2)*(t2-t1))/(x2-x1)                     #velocity of the frame s' (meter/second)

#Result
print"\nvelocity of the frame s =",v/(3*10**8),"c"

velocity of the frame s = -0.5 c


## Example no:6,Page no:33¶

In :
import numpy
#variable declaration
interval_s=1.0                                              #time difference between two events in frame s (second)
interval_sdash=4.0                                          #time difference between two events in frame s' (second)
separation_s=0.0                                           #spatial separation of two events in frame s (meter)
c=3.0*10**8                                                 #speed of light (meter/second)
v=numpy.random.rand()                                                   #assign a random value to unknown velocity(meter/second)
import math
#calculation
Gamma=interval_sdash/(interval_s-(v*(separation_s))/(c**2))  #calculating gamma
separation=-2.0*(((Gamma**2.0)-1)**(1.0/2.0))*c                      #spatial separation in s' (meter)

#Result
print"\nspatial separation of the events in frame s =",separation/(3*10**8*math.sqrt(15)),"c sqrt(15)"

spatial separation of the events in frame s = -2.0 c sqrt(15)


## Example no:7,Page no:33¶

In :
import numpy
#variable declaration
interval_s=0.0                                                   #time difference between two events in frame s (second)
separation_s=1.0                                                 #spatial separation of two events in frame s (meter)
separation_sdash=2.0                                             #spatial separation of two events in frame s' (meter)
c=3*10**8                                                       #speed of light (meter/second)
v=numpy.random.rand()                                                       #assign a random value to unknown velocity of frame s' with respect to frame s (meter/second)

#calculation
Gamma=separation_sdash/(separation_s-(v*interval_s))           #calculating value of Gamma
Beta=(1-1/(Gamma**2))**(1/2)                                     #calculating value of Beta
v=Beta*c                                                       #velocity of s' with respect to s (meter/second)
interval_sdash=Gamma*(interval_s-((v*separation_s)/(c**2)))     #time interval between the events in frame s' (second)

#Result
print"\nThe time interval between the events in frame s =",interval_sdash/(3*10**8),"X0"

The time interval between the events in frame s = -2.22222222222e-17 X0


## Example no:8,Page no:34¶

In :
#variable declaration
IbyI_not=.99                       #ratio of moving length and rest length
c=3*10**8                           #speed of light (meter/second)

#calculation
Beta=(1-IbyI_not**2)**(1/2.0)          #calculating value of Beta
v=Beta*c                           #velocity of rocket ship (meter/second)

#Result
print"\nThe velocity of the rocket ship = %.2e"%(v/(3*10**8)),"c"

The velocity of the rocket ship = 1.41e-01 c


## Example no:9,Page no:34¶

In :
#variable declaration
l_dash=1.0                                    #length of the rod in frame s' (meter)
Theta_dash_degree=45.0                        #angle of the rod with x-axis in frame s' (degree)
Beta=1/2.0                                    #value of Beta
import math

#calculation
tan_theta=math.tan(Theta_dash_radian)/((1.0-Beta**2)**(1.0/2.0))                              #tan of angle of rod with x-axis in frame s
theta=math.atan(tan_theta)                                                      #angle of rod with x-axis in frame s (degree)

#Result
print"\nThe length of the rod =",round(l,2),"m\nInclination of rod with x-axis =",round(math.degrees(theta))," degree"

The length of the rod = 0.94 m
Inclination of rod with x-axis = 49.0  degree


## Example no:10,Page no:34¶

In :
#variable declaration
c=3*10**8                                   #speed of light (meter/second)

#calculation
Beta=(1-((1/1.25)**2))**(1.0/2.0)                  #calculating Beta (1.25 comes from the fact that in frame s' density of bloc is 25% greater than frame s)
v=Beta*c                                     #velocity of the reference frame s'

#Result
print"\nBeta =",Beta
print"NOTE solved in book:\nThe velocity of the frame s = %.1e"%v," m/s"

Beta = 0.6
NOTE solved in book:
The velocity of the frame s = 1.8e+08  m/s


## Example no:11,Page no:35¶

In :
#Variable declaration
del_tao=1436.0     #min
del_t=1440.0       #min

#Calculation
def f(b):
return(del_t-del_tao/(math.sqrt(1-b**2)))

from scipy.optimize import fsolve
be=fsolve(f,0.5)

#Result
print"Beta=",be,"=1/(sqrt(180))"

Beta= 0.0744838204322 =1/(sqrt(180))


## Example no:12,Page no:35¶

In :
#variable declaration
deltaTow=1*10**(-6)                           #mean proper lifetime of particle (second)
Beta=0.9                                     #value of Beta
v=2.7*10**8                                   #velocity of particle (meter/second)

#Calculation
#part(i)
deltaT=deltaTow/((1-Beta**2)**(1.0/2.0))           #lifetime of the particle in the laboratory frame (second)
#part(ii)
d=v*deltaT                                   #distance traversed by the particle in the laboratory before disintegration (meter)

#Result
print"\nIn laboratory frame:\n\t(i)Lifetime of the particle = %.2e"%deltaT,"s\n"
print"\t(ii)Distance traversed by the particle = %.2g"%d," m"

In laboratory frame:
(i)Lifetime of the particle = 2.29e-06 s

(ii)Distance traversed by the particle = 6.2e+02  m


## Example no:13,Page no:35¶

In :
#variable declaration
d=3.0    #km
d=d*1000.0  #[m]
c=3.0*10**8  #m/s speed of light

#Calculation
v=0.99*c     #muon velocity
b=(v**2)/(c**2)

del_t=d/v
del_tao=del_t*math.sqrt(1-0.99**2)
#In moun's frame,
d_dash=d*math.sqrt(1-0.99**2)

#Result
print"(i)Proper lifetime of the muon= %.1e"%del_tao,"s"
print"(ii)In muon's frame,distance travelled by it is %.3e"%d_dash,"m"

(i)Proper lifetime of the muon= 1.4e-06 s
(ii)In muon's frame,distance travelled by it is 4.232e+02 m


## Example no:14,Page no:36¶

In :
#variable declaration
import sympy
c=sympy.Symbol("c")
u1=0.6*c                                  #speed of Beta particle 1 in lab frame (meter/second)
u2=-0.8*c                                 #speed of Beta particle 2 in lab frame (meter/second)
v=u1                                      #velocity of frame s' where frame s' is attached to the first Beta particle (meter/second)

#Calculation

u2_dash=(u2-v)/(1-((u2*v)/c**2))           #velocity of 2nd Beta particle relative to the 1st Beta particle (meter/second)

#Result
print"The velocity of 2nd Beta particle relative to the 1st Beta particle =",round(u2_dash/c,3)*c

The velocity of 2nd Beta particle relative to the 1st Beta particle = -0.946*c


## Example no:15,Page no:36¶

In :
#variable declaration
m0=1.0                                    #let rest mass of particle to be 1 (kg)
m=3.0*m0                                  #moving mass of particle (kg)
import sympy                               #speed of light (meter/second)
c=sympy.Symbol("c")

#calculation
Beta=(1-(m0/m)**2)**(1.0/2.0)                 #Calculation fo Beta
v=Beta*c                                #speed of particle (meter/second)

#Result
print"The speed of The particle =",round(v/c,3)*c,"=((2*sqrt 2)/3)c"

The speed of The particle = 0.943*c =((2*sqrt 2)/3)c


## Example no:23,Page no:38¶

In :
#Variable declaration
RestEnergy=0.51                            #energy of electron if the electron is at rest (Mev)
T=2                            #kinetic energy of electron (Bev)

#Calculation
#E=T=pc

from sympy import Symbol
c=Symbol("c")                               #speed of light (meter/second)
p=(T/c)                          #momentum of electron neglecting rest energy relative to kinetic energy (Bev*second/meter)

#Result
print"The momentum of the electron =",p,"BeV/c"

The momentum of the electron = 2/c BeV/c


## Example no:24,Page no:38¶

In :
#Variable declaration
n=0.01                            #fractional increase in momentum
c=3*10**8                          #speed of light (meter/second)

#Calculation
Beta=(n*(2-n))**(1.0/2.0)              #calculation of Beta
v=Beta*c                          #velocity of particle (meter/second)

#Result
print"\nBeta =",round(Beta,2)

Beta = 0.14


## Example no:26,Page no:39¶

In :
#Variable declaration
RestEnergy=0.51                                   #rest energy of electron (Mev)
T=1.0                                               #potential difference i.e. kinetic energy (Mev)
c=3*10**8                                          #speed of light (meter/second)

#Calculation
Beta=(1-(RestEnergy/(T+RestEnergy))**2)**(1.0/2.0)      #calculation of Beta
v=Beta*c                                          #speed of electron (meter/second)

#Result
print"The speed of the electron,Beta =",round(Beta,4)
print"Note: In the book answer of Beta is wrong"

The speed of the electron,Beta = 0.9412
Note: In the book answer of Beta is wrong


## Example no:27,Page no:39¶

In :
#Variable declaration
RestEnergy=0.51                                   #rest energy of electron (Mev)
T=2000.0                                            #potential difference i.e. kinetic energy (Mev)

#Calculation
import math
#part(i)effective mass of electron in terms of its rest mass
EffectiveMass=1+(T/RestEnergy)                    #ratio of effective mass of electron and rest mass
#part(ii)speed of electron in terms of the speed of light
Beta=(1-(1/EffectiveMass)**2)**(1.0/2.0)                #Calculatio of Beta
import sympy
eff_mass=sympy.Symbol("3923")
beta=((1-(1/eff_mass)**2))**(1.0/2.0)               #Calculatio of Beta

#Result
print"The effective mass of electron in terms of its rest mass is",round(EffectiveMass)
print"The speed of electron =",beta,"c is speed of light"
print"OR after solving:"
print"Speed of electron=%.2f"%Beta,"c"

The effective mass of electron in terms of its rest mass is 3923.0
The speed of electron = (1 - 1/3923**2)**0.5 c is speed of light
OR after solving:
Speed of electron=1.00 c



## Example no:28,Page no:39¶

In :
#Variable declaration
c=3*10**8                                                           #speed of light(meter/second)
v1=0.6*c                                                          #initial velocity of particle (meter/second)
v2=0.8*c                                                          #final velocity of particle (meter/second)

#Calculation
#Classically
W_Classic=0.5*((v2/c)**2-(v1/c)**2)                                 #ratio of work and m0*c**2 (mo is the rest mass of particle and c is the speed of light)
#Relativistically
W_Relative=(1/(1-(v2/c)**2)**(1.0/2.0))-(1/(1-(v1/c)**2)**(1.0/2.0))          #ratio of work and m0*c**2 (mo is the rest mass of particle and c is the speed of light)

#Result
print"\nWork required:\n\t Classically: Work =",W_Classic,"*m0*c**2\n\t Relativistically: Work =",round(W_Relative,3),"*m0*c**2\nWhere m0:rest mass of particle & c:speed of light"

Work required:
Classically: Work = 0.14 *m0*c**2
Relativistically: Work = 0.417 *m0*c**2
Where m0:rest mass of particle & c:speed of light


## Example no:29,Page no:40¶

In :
#Variable declaration
h=6.63*10**-34                                     #planck's constant (joule*second)
c=3*10**8                                          #speed of light (meter/second)
lambda1=5000*10**-10                               #wavelength (meter)
lambda2=0.1*10**-10                                #wavelength (meter)

#Calculation
#part(i): wavelength=5000 Å
m1=h/(lambda1*c)                                  #effective mass of photon of wavelength 5000 Å
#part(ii): wavelength=0.1 Å
m2=h/(lambda2*c)                                  #effective mass of photon of wavelength 0.1 Å

#Result
print"Effective mass of photon:\n\t(i) mass =",m1,"kg\n\t(ii) mass =",m2,"kg"

Effective mass of photon:
(i) mass = 4.42e-36 kg
(ii) mass = 2.21e-31 kg


## Example no:30,Page no:40¶

In :
#Variable declaration
RestEnergy=0.51                                   #rest energy of electron (Mev)

#Calculation
E=2*RestEnergy                                    #minimum energy of gamma ray photon (Mev)

#Result
print"\nMinimum energy required =",E,"Mev"

Minimum energy required = 1.02 Mev


## Example no:33,Page no:41¶

In :
#Variable declaration
c=3*10**8                                       #Speed of sound (meter/second)
M=1.97*10**30                                   #Mass of sun (kg)
R=1.5*10**11                                    #Mean radius of the earth orbit (meter)
sigma=1.4*10**3                                 #Solar energy received by the earth (joule/meter**2*second)

#Calculation
import math
loss=(4*math.pi*R**2*sigma)/(M*c**2)                 #Fractional loss of mass of the sun per second

#Result
print"\nThe fractional loss of mass of the sun= %.e"%loss,"s**-1"

The fractional loss of mass of the sun= 2e-21 s**-1