Introduction To Special Relativity And Space Science (By S.P. Singh)

CHAPTER NUMBER 6 : RELATIVISTIC KINEMATICS AND PARADOXES IN RELATIVITY

EXAMPLE NUMBER 6.1 : (Page Number 221)

In [102]:
import math
   #Given that
c = 3e8    # speed of light in m/s
v = 0.8* c    # velocity of rod
l1 = 1    # let 
theta1 = 60    # anlge between length of rod and speed in degree
l_x = l1 * math.cos(theta1 * pi /180) * math.sqrt (1-(v /c)**2)
l_y = l1 * math.sin(theta1 * pi /180)
l2 = math.sqrt (l_x**2 + l_y**2)
per_contrtaction = (l1 - l2) / l1 *100
angle = math.atan (l_y/l_x)
print "Percentage contraction in rod is ",round(per_contrtaction,4)," and apparant orientation is ",round(math.tan(angle),4),""

Percentage contraction in rod is  8.3645  and apparant orientation is  2.8832

EXAMPLE NUMBER 6.2 : (Page Number 222)

In [108]:
import math
   #Given that
c = 3e8    # speed of light in m/s
u_x_ = -3e8    # velocity of first photon in ground frame in m/s
v = -3e8    # velocity of second photon in ground frame in m/s
print "Standard formula used is  u_x = (u_x_ + v) / (1 + v * u_x_ / c**2) "
u_x = (u_x_ + v) / (1 + v * u_x_ / c**2)    # calculation of Velocity of photon with respect to another
print "Velocity of photon with respect to another is ",(u_x / c)," * c   Thus photons are approaching each other."

Standard formula used is  u_x = (u_x_ + v) / (1 + v * u_x_ / c**2) 
Velocity of photon with respect to another is  -1.0  * c   Thus photons are approaching each other.

EXAMPLE NUMBER 6.3 : (Page Number 222)

In [111]:
import math
   #Given that
c = 3e8    # speed of light in m/s
u_x_ = -0.9 * c    # velocity of first spaceship in ground frame in m/s
v = -0.9 *c    # velocity of second spaceship in ground frame in m/s
print "Standard formula used is  u_x = (u_x_ + v) / (1 + v * u_x_ / c**2) "
u_x = (u_x_ + v) / (1 + v * u_x_ / c**2)    # calculation of Velocity of photon
print "Velocity of photon with respect to another is ",round(u_x / c,4)," c."

Standard formula used is  u_x = (u_x_ + v) / (1 + v * u_x_ / c**2) 
Velocity of photon with respect to another is  -0.9945  c.

EXAMPLE NUMBER 6.4 : (Page Number 223)

In [112]:
import math
   #Given that
E = 7.5e11    # Energy in kWh
c = 3e8    # speed of light in m/s
print "  Standard formula used   E = m*c**2"
m = (E *3.6e6) / c**2   # calculation of Amount of mass consumed
print "  Amount of mass consumed is ",m," kg."

  Standard formula used   E = m*c**2
  Amount of mass consumed is  30.0  kg.

EXAMPLE NUMBER 6.5 : (Page Number 223)

In [113]:
import math
   #Given that
m = 4    # mass of substance consumed fully in kg
c = 3e8    # speed of light in m/s
print "  Standard formula used   E = m*c**2"
E = m * c**2   # calculation of Amount of energy produced
print "  Amount of energy produced is ",E," J."

  Standard formula used   E = m*c**2
  Amount of energy produced is  3.6e+17  J.

EXAMPLE NUMBER 6.6 : (Page Number 223)

In [114]:
import math
   #Given that
m_0 = 1e-24    # mass of moving particle in kg
v = 1.8e8    # speed of particle in m/s
c = 3e8    # speed of light in m/s
print "  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)"
m = m_0 / math.sqrt(1 - (v / c)**2)    # calculation of Relativistic mass of particle
print "  Relativistic mass of particle is ",m," kg."

  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)
  Relativistic mass of particle is  1.25e-24  kg.

EXAMPLE NUMBER 6.7 : (Page Number 223)

In [116]:
import math
   #Given that
c = 3e8    # speed of light in m/s
v = 0.5 * c    # speed of particle in m/s

print "  Standard formula used   m = m_o/math.sqrt ( 1- (v/c)**2)"
ratio = math.sqrt(1- (v /c)**2)    # calculation of Ratio of rest mass and relativistic mass of particle
print "  Ratio of rest mass and relativistic mass of particle is ",round(ratio,4),"."

  Standard formula used   m = m_o/math.sqrt ( 1- (v/c)**2)
  Ratio of rest mass and relativistic mass of particle is  0.866 .

EXAMPLE NUMBER 6.8a : (Page Number 224)

In [119]:
import math
   #Given that
ratio = 0.5    # Ratio of lengths of spaceship
c = 3e8    # speed of light in m/s
print "Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)"
v = c * math.sqrt(1 - ratio**2)    # calculation of Speed of spaceship
print "  Speed of spaceship is ",v," m/s."

  Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)
  Speed of spaceship is  259807621.135  m/s.

EXAMPLE NUMBER 6.8b : (Page Number 224)

In [121]:
import math
   #Given that
c = 3e8    # speed of light in m/s
v = 2.598e8    # speed of spaceship
t_0 = 1    # time in second
print "  Standard formula used   t= t_o/ math.sqrt ( 1- (v/c)**2)"
t = t_0 / math.sqrt(1 - (v **2 / c **2) )    # calculation of Time corresponding to 1 sec
print "  Time corresponding to 1 sec is ",math.ceil (t)," sec."

   Standard formula used   t= t_o/ math.sqrt ( 1- (v/c)**2)
  Time corresponding to 1 sec is  2.0  sec.

EXAMPLE NUMBER 6.9 : (Page Number 224)

In [122]:
import math
   #Given that
c = 3e8    # speed of light in m/s
v = 2.4e8    # speed of meson
t_0 = 2e-8    # lifetime of meson in second
print "  Standard formula used "
t = t_0 / math.sqrt(1 - (v  / c )**2 )    # calculation of Lifetime of meson
print "  Lifetime of meson is ",t," sec."

  Standard formula used 
  Lifetime of meson is  3.33333333333e-08  sec.

EXAMPLE NUMBER 6.10 : (Page Number 225)

In [123]:
import math
   #Given that
c = 3e8    # speed of light in m/s
m_0 = 1    # atomic mass in amu
m = 3 * m_0    # relativistic mass
print "  Standard formula used  l = l_o * math.sqrt ( 1- (v/c)**2)"
v = c * math.sqrt(1- (m_0 / m)**2)    # calculation of Velocity of particle
print "  Velocity of particle is ",v / c," c."

  Standard formula used  l = l_o * math.sqrt ( 1- (v/c)**2)
  Velocity of particle is  1.0  c.

EXAMPLE NUMBER 6.11 : (Page Number 225)

In [124]:
import math
   #Given that
mass_ratio = 0.5    # Ratio of rest mass and relativistic mass  
c = 3e8    # speed of light in m/s
print "  Standard formula used   m = m_o / math.sqrt ( 1- (v/c)**2) "
v = c * math.sqrt(1- mass_ratio**2)    # calculation of Velocity of particle
print "  Velocity of particle is ",v / c," c."

  Standard formula used   m = m_o / math.sqrt ( 1- (v/c)**2) 
  Velocity of particle is  0.866025403784  c.

EXAMPLE NUMBER 6.12a : (Page Number 226)

In [125]:
import math
   #Given that
c = 3e8    # speed of light in m/s
u_x_ = -2e8    # velocity of first photon in ground frame in m/s
v = -2e8    # velocity of second photon in ground frame in m/s
m_0 = 3e-25
print "  Standard formula used   u_x = (u_x_ + v) / (1 + v * u_x_ / c**2)"
u_x = (u_x_ + v) / (1 + v * u_x_ / c**2)    # calculation of Velocity of photon with respect to another
m = m_0 / math.sqrt(1 - (u_x / c)**2)    # calculation of Relativistic mass of particle with respect to another
print "  Velocity of photon with respect to another is ",u_x," m/s."
print "  Relativistic mass of particle with respect to another is ",m," kg."

  Standard formula used   u_x = (u_x_ + v) / (1 + v * u_x_ / c**2)
  Velocity of photon with respect to another is  -276923076.923  m/s.
  Relativistic mass of particle with respect to another is  7.8e-25  kg.

EXAMPLE NUMBER 6.12b : (Page Number 227)

In [126]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
ratio = 1.95e+03    # Ratio of relativistic mass and rest mass
print "  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)"
ratio_1 = 1 /(2* ratio**2)       # calculation of ratio of velocity to velocity of light for	
print "  Ratio of velocity to velocity of light for particle is 1 -  ",ratio_1," ."

  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)
  Ratio of velocity to velocity of light for particle is 1 -   1.31492439185e-07  .

EXAMPLE NUMBER 6.13 : (Page Number 226)

In [127]:
import math
   #Given that
c = 3e8    # speed of light in m/s
u = 0.9*c    # velocity of first particle with respect to other in m/s
density1 = 19.3e-3    # density of gold in rest frame
print "  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)   and   l = l_o* math.sqrt ( 1- (v/c)**2) "
mass_ratio = math.sqrt (1 - (u/c)**2)    # calculation of ratio of relativistic mass
volume_ratio = 1 / math.sqrt (1 - (u/ c)**2)    # calculation of ratio of relativistic volume
density2 = density1 * (volume_ratio /mass_ratio )    #calculation of ratio of relativistic density
print "  Relativistic density of rod in moving frame is ",density2,"."

  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)   and   l = l_o* math.sqrt ( 1- (v/c)**2) 
  Relativistic density of rod in moving frame is  0.101578947368 .

EXAMPLE NUMBER 6.14a : (Page Number 227)

In [128]:
import math
   #Given that
E = 1e9    # energy of electron in eV  
c = 3e8    # speed of light in m/s
m_0 = 9.1e-31    # mass of electron in kg
print "  Standard formula used    E = m*c**2"
m = E / c**2 * 1.6e-19         # calculation of relativistic mass of particle
ratio = m / m_0   # calculation of Ratio of relativistic mass and rest mass of particle
print "  Ratio of relativistic mass and rest mass  of particle is ",ratio,"."

  Standard formula used    E = m*c**2
  Ratio of relativistic mass and rest mass  of particle is  1953.6019536 .

EXAMPLE NUMBER 6.14b : (Page Number 227)

In [129]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
ratio = 1.95e+03    # Ratio of relativistic mass and rest mass
print "  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)"
ratio_1 = 1 /(2* ratio**2)       # calculation of ratio of velocity to velocity of light for	
print "  Ratio of velocity to velocity of light for particle is 1 -  ",ratio_1," ."

  Standard formula used   m = m_o/ math.sqrt ( 1- (v/c)**2)
  Ratio of velocity to velocity of light for particle is 1 -   1.31492439185e-07  .

EXAMPLE NUMBER 6.14c : (Page Number 227)

In [130]:
import math
   #Given that 
m = 9e-31    # mass in kg
E = 1e9    # Energy of accelerated electron in eV
c = 3e8    # speed of light in m/s
print "  Standard formula used   E = m*c**2"
E_0 = m * c**2   # calculation of rest mass energy
ratio = E / E_0 *1.6e-19   # calculation of Ratio of energy to rest mass energy
print "  Ratio of energy to rest mass energy is ",ratio,"."

  Standard formula used   E = m*c**2
  Ratio of energy to rest mass energy is  1975.30864198 .

EXAMPLE NUMBER 6.15 : (Page Number 228)

In [133]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
v = 0.6 * c    # velocity of rod wrt laboratory
l_ = 1    # length of rod measured by observer in lab
print "  Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)"
l = l_ / math.sqrt (1 - (v / c)**2)     # calculation of Proper length of rod  
print "  Proper length of rod  is ",l," m."

  Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)
  Proper length of rod  is  1.25  m.

EXAMPLE NUMBER 6.16 : (Page Number 228)

In [134]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
v = 0.9 * c    # velocity of rod wrt laboratory
proper_time = 2.5e-8    # proper mean life time of mesons
print "  Standard formula used   t = t_o /math.sqrt ( 1- (v/c)**2)"
t = proper_time / math.sqrt (1 - (v / c)**2)    # calculation of New mean life time
print "  New mean life time is ",t," s."

  Standard formula used   t = t_o /math.sqrt ( 1- (v/c)**2)
  New mean life time is  5.73539334676e-08  s.

EXAMPLE NUMBER 6.17 : (Page Number 229)

In [135]:
import math
   #Given that
E = 1    # energy of electron in MeV  
c = 3e8    # speed of light in m/s
m_0 = 9e-31    # rest mass of electron
print "  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2)   and   E=m*c**2"
m = E * 1.6e-13 / c**2   # calculation of mass of electron
v = c * math.sqrt(1 - (m_0 / m)**2)    # calculation of Velocity of electron
print "  Velocity of electron is ",v," m/s."

  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2)   and   E=m*c**2
  Velocity of electron is  258716030.379  m/s.

EXAMPLE NUMBER 6.19 : (Page Number 230)

In [136]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
v = 0.99 * c     # velocity of particle
proper_time = 2.2e-6    # proper mean life time of mesons
print "  Standard formula used   t = t_o /math.sqrt ( 1- (v/c)**2)"
t = proper_time / math.sqrt (1 - (v / c)**2)    # calculation of time period
d = v *t   # calculation of Distance travelled by particle
print "  Distance traveled by particle is ",d," m."

  Standard formula used   t = t_o /math.sqrt ( 1- (v/c)**2)
  Distance traveled by particle is  4631.82979352  m.

EXAMPLE NUMBER 6.20 : (Page Number 230)

In [137]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
m = 1    # let 
m_change = 1    # change in mass in percentage by increamath.sing velocity
print "  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2) "
v = c * math.sqrt (1 - (m / (m + m_change/100))**2)    # calculation of Velocity required to increase mass by one percent
print  "  Velocity required to increase mass by one perfect is ",v," m/s."

  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2) 
  Velocity required to increase mass by one perfect is  0.0  m/s.

EXAMPLE NUMBER 6.21 : (Page Number 231)

In [138]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
m_ratio = 2000    # ratio of rest mass and relativistic mass
print "  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2) "
v = c * math.sqrt (1 - (1/m_ratio)**2)    # calculation of Velocity required to increase mass by 2000 times
	
print  "  Velocity required to increase mass by 2000 times is ",c," - ",(c -v)," m/s."

  Standard formula used   m = m_o* math.sqrt ( 1- (v/c)**2) 
  Velocity required to increase mass by 2000 times is  300000000.0  -  0.0  m/s.

EXAMPLE NUMBER 6.22 : (Page Number 231)

In [139]:
import math
   #Given that 
h = 6.63e-34    # plank's constant
c = 3e8    # speed of light in m/s
lambda1 = 5e-4    # wavelength of photon in angstrom
e_rest_mass = 0.511    # rest mass of electron in Mev/c**2
p_rest_mass = 0.511    # rest mass of electron in Mev/c**2
print "  Standard formula used   E_total = E_rest + E_kinetic"
k = (((h * c  / (lambda1 * 1.6e-23 ))  - (e_rest_mass + p_rest_mass))) /2    # calculation of Energy of each particle
print  "  Energy of each particle is  ",k," MeV."   

  Standard formula used   E_total = E_rest + E_kinetic
  Energy of each particle is   11.92025  MeV.

EXAMPLE NUMBER 6.23 : (Page Number 232)

In [141]:
import math
   #Given that 
h = 6.63e-34    # plank's constant
c = 3e8    # speed of light in m/s
p_rest_mass = 938    # rest mass of proton in Mev/
ap_rest_mass = 938    # rest mass of antiproton in Mev
print "  Standard formula used   E = h* c / lambda1"
lambda1 = h * c / ((p_rest_mass + ap_rest_mass) * 1.6e-19)    # calculation of  Threshold wavelength for proton - antiproton production
print  "  Threshold wavelength for proton - antiproton production is  ",round((lambda1 / 1e-10),4)," angstrom."

  Standard formula used   E = h* c / lambda1
  Threshold wavelength for proton - antiproton production is   6.6265  angstrom.

EXAMPLE NUMBER 6.24 : (Page Number 232)

In [142]:
import math
   #Given that
c = 3e8    # speed of light in m/s
p_rest_mass = 0.938    # rest mass energy of proton in BeV
KE = 1    # kinetic energy of proton in BeV 
print "  Standard formula used E**2 = p**2*c**2 + m_o**2*c**4*"
E = KE + p_rest_mass   # calculation of energy of particle
p = (math.sqrt (E**2 *1e6 - (p_rest_mass * 1e3)**2)) *(1.6e-19)*(1e9) / c   # calculation of Momentum of photon
print  "  Momentum of photon is ",p," kg m/s."

  Standard formula used E**2 = p**2*c**2 + m_o**2*c**4*
  Momentum of photon is  9.04467922163e-16  kg m/s.

EXAMPLE NUMBER 6.26 : (Page Number 228)

In [144]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
t = 8e-6    # mean life of meson 
l = 10    # distance of meson from earth surface
print "  Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)"
v = l*1e3/ math.sqrt( t**2 +(l * 1e3 /c)**2)    # calculation of relative speed of meson with respect to
print "  Relative speed of meson with respect to earth  is ",round(v/c,4)," c ."

  Standard formula used   l = l_o* math.sqrt ( 1- (v/c)**2)
  Relative speed of meson with respect to earth  is  0.9724  c .

EXAMPLE NUMBER 6.27 : (Page Number 228)

In [145]:
import math
   #Given that 
c = 3e8    # speed of light in m/s
v = 0.8 *c    # velocity of rod in m/s
m_0 = 1.673e-27    # rest mass of proton in kg
print "  Standard formula used   E_total = KE + E_mass"
K_E = m_0 * c**2 *(1/math.sqrt(1-(v/c)**2) - 1) / 1.6e-13    # calculation of Kinetic energy of proton
print "  Kinetic energy of proton is ",K_E,"MeV."

  Standard formula used   E_total = KE + E_mass
  Kinetic energy of proton is  627.375 MeV.