8: Special Theory of Relativity

Example number 8.1, Page number 171

In [1]:
#importing modules
import math
from __future__ import division

#Variable declaration
L_0 = 1;     #For simplicity, we assume classical length to be unity(m)
c = 1;       #For simplicity assume speed of light to be unity(m/s)

#Calculation
L = (1-1/100)*L_0;     #Relativistic length(m)
#Relativistic length contraction gives L = L_0*sqrt(1-v^2/c^2), solving for v
v = math.sqrt(1-(L/L_0)**2)*c;    #Speed at which relativistic length is 1 percent of the classical length(m/s)
v = math.ceil(v*10**4)/10**4;     #rounding off the value of v to 4 decimals

#Result
print "The speed at which relativistic length is 1 percent of the classical length is",v, "c"

The speed at which relativistic length is 1 percent of the classical length is 0.1411 c

Example number 8.2, Page number 171

In [2]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;      #For simplicity assume speed of light to be unity(m/s)
delta_t = 5*10**-6;    #Mean lifetime of particles as observed in the lab frame(s)

#Calculation
v = 0.9*c;    #Speed at which beam of particles travel(m/s)
delta_tau = delta_t*math.sqrt(1-(v/c)**2);     #Proper lifetime of particle as per Time Dilation rule(s)

#Result
print "The proper lifetime of particle is",delta_tau, "s"

The proper lifetime of particle is 2.17944947177e-06 s

Example number 8.3, Page number 171. theoritical proof

Example number 8.4, Page number 172

In [4]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;      #For simplicity assume speed of light to be unity(m/s)

#Calculation
v = 0.6*c;    #Speed with which the rocket leaves the earth(m/s)
u_prime = 0.9*c;     #Relative speed of second rocket w.r.t. the first rocket(m/s)
u1 = (u_prime+v)/(1+(u_prime*v)/c**2);     #Speed of second rocket for same direction of firing as per Velocity Addition Rule(m/s)
u1 = math.ceil(u1*10**4)/10**4;     #rounding off the value of u1 to 4 decimals
u2 = (-u_prime+v)/(1-(u_prime*v)/c**2);     #Speed of second rocket for opposite direction of firing as per Velocity Addition Rule(m/s)
u2 = math.ceil(u2*10**4)/10**4;     #rounding off the value of u2 to 4 decimals

#Result
print "The speed of second rocket for same direction of firing is",u1,"c"
print "The speed of second rocket for opposite direction of firing is",u2,"c"

The speed of second rocket for same direction of firing is 0.9741 c
The speed of second rocket for opposite direction of firing is -0.6521 c

Example number 8.5, Page number 172

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;     #For simplicity assume speed of light to be unity(m/s)
L0 = 1;    #For simplicity assume length in spaceship's frame to be unity(m)
tau = 1;     #Unit time in the spaceship's frame(s)

#Calculation
L = 1/2*L0;    #Length as observed on earth(m)
#Relativistic length contraction gives L = L_0*sqrt(1-v^2/c^2), solving for v
v = math.sqrt(1-(L/L0)**2)*c;    #Speed at which length of spaceship is observed as half from the earth frame(m/s)
t = tau/math.sqrt(1-(v/c)**2);    #Time dilation of the spaceship's unit time(s)
v = math.ceil(v*10**4)/10**4;     #rounding off the value of v to 4 decimals

#Result
print "The speed at which length of spaceship is observed as half from the earth frame is",v, "c"
print "The time dilation of the spaceship unit time is",t,"delta_tau"

The speed at which length of spaceship is observed as half from the earth frame is 0.8661 c
The time dilation of the spaceship unit time is 2.0 delta_tau

Example number 8.6, Page number 172

In [6]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;     #Speed of light in vacuum(m/s)
t1 = 2*10**-7;      #Time for which first event occurs(s)
t2 = 3*10**-7;      #Time for which second event occurs(s)
x1 = 10;       #Position at which first event occurs(m)
x2 = 40;       #Position at which second event occurs(m)

#Calculation
v = 0.6*c;       #Velocity with which S2 frame moves relative to S1 frame(m/s)
L_factor = 1/math.sqrt(1-(v/c)**2);     #Lorentz factor
delta_t = L_factor*(t2 - t1)+L_factor*v/c**2*(x1 - x2);     #Time difference between the events(s)
delta_x = L_factor*(x2 - x1)-L_factor*v*(t2 - t1);       #Distance between the events(m)

#Result
print "The time difference between the events is",delta_t, "s" 
print "The distance between the events is",delta_x, "m"

The time difference between the events is 5e-08 s
The distance between the events is 15.0 m

Example number 8.7, Page number 173

In [9]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;     #Speed of light in vacuum(m/s)
tau = 2.6*10**-8;     #Mean lifetime the particle in its own frame(s)
d = 20;     #Distance which the unstable particle travels before decaying(m)

#Calculation
#As t = d/v and also t = tau/sqrt(1-(v/c)^2), so that
#d/v = tau/sqrt(1-(v/c)^2), solving for v
v = math.sqrt(d**2/(tau**2+(d/c)**2));     #Speed of the unstable particle in lab frame(m/s)
v = v/10**8;
v = math.ceil(v*10)/10;     #rounding off the value of v to 1 decimal

#Result
print "The speed of the unstable particle in lab frame is",v,"*10**8 m/s"

The speed of the unstable particle in lab frame is 2.8 *10**8 m/s

Example number 8.8, Page number 174

In [13]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;     #For simplicity assume speed of light to be unity(m/s)
me = 1;    #For simplicity assume mass of electron to be unity(kg)
tau = 2.3*10**-6;     #Average lifetime of mu-meson in rest frame(s)
t = 6.9*10**-6;       #Average lifetime of mu-meson in laboratory frame(s)
e = 1.6*10**-19;     #Energy equivalent of 1 eV(J/eV)
C = 3*10**8;     #Speed of light in vacuum(m/s)
m_e = 9.1*10**-31;     #Mass of an electron(kg)

#Calculation
#Fromm Time Dilation Rule, tau = t*sqrt(1-(v/c)^2), solving for v
v = c*math.sqrt(1-(tau/t)**2);     #Speed of mu-meson in the laboratory frame(m/s)
v = math.ceil(v*10**5)/10**5;     #rounding off the value of v to 5 decimals
m0 = 207*me;     #Rest mass of mu-meson(kg)
m = m0/math.sqrt(1-(v/c)**2);      #Relativistic variation of mass with velocity(kg)
m = math.ceil(m*10)/10;     #rounding off the value of m to 1 decimal
T = (m*m_e*C**2 - m0*m_e*C**2)/e;     #Kinetic energy of mu-meson(eV)
T = T*10**-6;        #Kinetic energy of mu-meson(MeV)
T = math.ceil(T*100)/100;     #rounding off the value of T to 2 decimals
 
#Result
print "The speed of mu-meson in the laboratory frame is",v, "c"
print "The effective mass of mu-meson is",m, "me"
print "The kinetic energy of mu-meson is",T, "MeV"

The speed of mu-meson in the laboratory frame is 0.94281 c
The effective mass of mu-meson is 621.1 me
The kinetic energy of mu-meson is 211.97 MeV

Example number 8.9, Page number 174

In [14]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;      #For simplicity assume speed of light to be unity(m/s)
m0 = 1;     #For simplicity assume rest mass to be unity(kg)

#Calculation
m = (20/100+1)*m0;     #Mass in motion(kg)
#As m = m0/sqrt(1-(u/c)^2), solving for u
u = math.sqrt(1-(m0/m)**2)*c;     #Speed of moving mass(m/s) 
u = math.ceil(u*10**3)/10**3;     #rounding off the value of u to 3 decimals

#Result
print "The speed of moving body is",u, "c"

The speed of moving body is 0.553 c

Example number 8.10, Page number 175

In [18]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;     #Speed of light in vacuum(m/s)
dE = 4*10**26;     #Energy radiated per second my the sun(J/s)

#Calculation
dm = dE/c**2;       #Rate of decrease of mass of sun(kg/s)
dm = dm/10**9;
dm = math.ceil(dm*10**3)/10**3;     #rounding off the value of dm to 3 decimals

#Result
print "The rate of decrease of mass of sun is",dm,"*10**9 kg/s"

The rate of decrease of mass of sun is 4.445 *10**9 kg/s

Example number 8.11, Page number 175

In [19]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 1;     #For simplicity assume speed of light to be unity(m/s)
m0 = 9.1*10**-31;    #Mass of the electron(kg)
E0 = 0.512;         #Rest energy of electron(MeV)
T = 10;         #Kinetic energy of electron(MeV)

#Calculation
E = T + E0;     #Total energy of electron(MeV)
# From Relativistic mass-energy relation E^2 = c^2*p^2 + m0^2*c^4, solving for p
p = math.sqrt(E**2-m0**2*c**4)/c;      #Momentum of the electron(MeV)
p = math.ceil(p*100)/100;     #rounding off the value of p to 2 decimals
#As E = E0/sqrt(1-(u/c)^2), solving for u
u = math.sqrt(1-(E0/E)**2)*c;     #Velocity of the electron(m/s)
u = math.ceil(u*10**4)/10**4;     #rounding off the value of u to 4 decimals

#Result
print "The momentum of the electron is",p,"/c MeV"
print "The velocity of the electron is",u, "c"

#answer for velocity given in the book is wrong

The momentum of the electron is 10.52 /c MeV
The velocity of the electron is 0.9989 c

Example number 8.12, Page number 175. theoritical proof

Example number 8.13, Page number 176

In [20]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;      #Speed of light in vacuum(m/s)
E = 4.5*10**17;   #Total energy of object(J)
px = 3.8*10**8;    #X-component of momentum(kg-m/s)
py = 3*10**8;      #Y-component of momentum(kg-m/s)
pz = 3*10**8;      #Z-component of momentum(kg-m/s)

#Calculation
p = math.sqrt(px**2+py**2+pz**2);     #Total momentum of the object(kg-m/s)
#From Relativistic mass-energy relation E^2 = c^2*p^2 + m0^2*c^4, solving for m0
m0 = math.sqrt(E**2/c**4 - p**2/c**2);    #Rest mass of the body(kg)
m0 = math.ceil(m0*100)/100;     #rounding off the value of m0 to 2 decimals

#Result
print "The rest mass of the body is",m0, "kg"

The rest mass of the body is 4.63 kg

Example number 8.14, Page number 176

In [21]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;     #Speed of light in vacuum(m/s)
m = 50000;       #Mass of high speed probe(kg)

#Calculation
u = 0.8*c;       #Speed of the probe(m/s)
p = m*u/math.sqrt(1-(u/c)**2);     #Momentum of the probe(kg-m/s)

#Result
print "The momentum of the high speed probe is",p, "kg-m/s"

The momentum of the high speed probe is 2e+13 kg-m/s

Example number 8.15, Page number 177

In [24]:
#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;     #Electronic charge, C = Energy equivalent of 1 eV(J/eV)
m0 = 9.11*10**-31;   #Rest mass of electron(kg)
c = 3*10**8;     #Speed of light in vacuum(m/s)

#Calculation
u1 = 0.98*c;     #Inital speed of electron(m/s)
u2 = 0.99*c;     #Final speed of electron(m/s)
m1 = m0/math.sqrt(1-(u1/c)**2);    #Initial relativistic mass of electron(kg)
m2 = m0/math.sqrt(1-(u2/c)**2);    #Final relativistic mass of electron(kg)
dm = m2 - m1;     #Change in relativistic mass of the electron(kg)
W = dm*c**2/e;      #Work done on the electron to change its velocity(eV)
W = W*10**-6;      #Work done on the electron to change its velocity(MeV)
W = math.ceil(W*100)/100;     #rounding off the value of W to 2 decimals
#As W = eV, V = accelerating potential, solving for V
V = W*10**6;     #Accelerating potential(volt)
V = V/10**6;

#Result
print "The change in relativistic mass of the electron is",dm, "kg"
print "The work done on the electron to change its velocity is",W, "MeV"
print "The accelerating potential is",V, "*10**6 volt"

#answers given in the book are wrong

The change in relativistic mass of the electron is 1.87996052912e-30 kg
The work done on the electron to change its velocity is 1.06 MeV
The accelerating potential is 1.06 *10**6 volt
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