# Chapter 1: Relativistic Mechanics¶

## Example 1.2, Page 26¶

In [3]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
delta_x = 2.45e+03;    # Space difference, m
delta_t = 5.35e-06;    # Time difference, s

#Calculations
v = 0.855*c;    # Speed of frame S_prime, m/s
delta_x_prime = 1/sqrt(1-v**2/c**2)*(delta_x - v*(delta_t))*1e-03;    # Distance between two flashes as measured in S_prime frame, km
delta_t_prime = 1/sqrt(1-v**2/c**2)*(delta_t - v/c**2*delta_x)*1e+006;    # Time between two flashes as measured in S_prime

#Results
print "The distance between two flashes as measured in S_prime frame = %4.2f km"%delta_x_prime
print "The time between two flashes as measured in S_prime frame = %4.2f micro-second"%delta_t_prime

The distance between two flashes as measured in S_prime frame = 2.08 km
The time between two flashes as measured in S_prime frame = -3.15 micro-second


## Example 1.4, Page 27¶

In [8]:
from sympy import *
c = Symbol('c')

#Variable declaration
c = 1;            # Speed of light in vacuum, m/s
u_x_prime = c;    # Velocity of photon as measured in S_prime frame, m/s
v = c;            # Velocity of frame S_prime relative to S frame, m/s

#Calculations
u_x = (u_x_prime + v)/(1+v*u_x_prime/c**2);
if u_x == 1:
ux = 'c';
else:
ux = string(u_x)+'c';

#Result
print "The speed of one photon as observed by the other is %c"%ux

The speed of one photon as observed by the other is c


## Example 1.6, Page 28¶

In [3]:
import math

#Variable declaration
a = 1;               # For simplicity assume length of semi minor axis to be unity, m
c = 3e+08;           # Speed of light, m/s

#Calculations
#From equation 1-v^2/c^2=1/4, we derive the following expression
v = math.sqrt(3*c**2/4)    # Velocity at which surface area of lamina reduces to half in S-frame, m/s

print "The velocity at which surface area of lamina reduces to half in S-frame = %4.2e m/s"%v

The velocity at which surface area of lamina reduces to half in S-frame = 2.60e+08 m/s


## Example 1.7, Page 29¶

In [13]:
#Variable declaration
m0 = 1;    # For simplicity assume the rest mass of stick to be unity, kg
m = 1.5*m0;    # Mass of the moving stick, kg
L0 = 1;    # Assume resting length of the stick to be unity, m

#Calculations
# As m = m0/sqrt(1-v^2/c^2) = m0*gama, solving for gama
gama = m/m0;    # Relativistic factor
L = L0/gama;    # Contracted length of the metre stick, m

#Result
print "The contracted length of the metre stick = %4.2f m"%L


The contracted length of the metre stick = 0.67 m


## Example 1.8, Page 29¶

In [14]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
tau0 = 2e-008;    # Mean lifetime of meson at rest, m/s

#Calculations
v = 0.8*c;    # Velocity of moving meason, m/s
tau = tau0/sqrt(1-v**2/c**2);    # Mean lifetime of meson in motion, m/s

#Result
print "The mean lifetime of meson in motion = %4.2e s"%tau

The mean lifetime of meson in motion = 3.33e-08 s


## Example 1.9, Page 30¶

In [8]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
delta_t0 = 59;    # Reading of the moving clock for each hour, min
delta_t = 60;    # Reading of the stationary clock for each hour, min

#Calculations
# As from Time Dilation, delta_t = delta_t0/sqrt(1-v^2/c^2), solving for v
v = sqrt(((delta_t**2-delta_t0**2)*c**2)/delta_t**2)

#Result
print "The speed at which the moving clock ticks slow = %4.2e m/s"%v

The speed at which the moving clock ticks slow = 5.45e+07 m/s


## Example 1.10, Page 30¶

In [17]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
tau0 = 2.5e-008;    # Mean lifetime of meson at rest, m/s

#Calculations
v = 0.8*c;    # Velocity of moving meason, m/s
tau = tau0/sqrt(1-v**2/c**2);    # Mean lifetime of meson in motion, m/s
N0 = 1;    # Assume initial flux of meson beam to be unity, watt/Sq.m
N = N0*exp(-2);    # Meson flux after time t, watt/Sq.m
# As N = N0*e^(-t/tau), which on comparing gives
t = 2*tau;    # Time during which the meson beam flux reduces, s
d = 0.8*c*t;    # The distance that the meson beam can travel before reduction in its flux, m

#Result
print "The distance that the meson beam can travel before reduction in its flux = %2d m"%d

The distance that the meson beam can travel before reduction in its flux = 20 m


## Example 1.11, Page 31¶

In [14]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
E0 = 1;    # Rest energy of particle, unit

#Calculations
E = 3*E0;    # Energy of relativistically moving particle, unit
# E = m*c^2 and E0 = m0*c^2
# With m = m0/sqrt(1-v^2/c^2), we have
v = c*sqrt(1-(E0/E)**2);    # Velocity of the moving particle, m/s

#Result
print "The velocity of the moving particle = %4.2e m/s"%v
#answer differs due to rounding-off errors

The velocity of the moving particle = 3.00e+08 m/s


## Example 1.12, Page 32¶

In [19]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
m0 = 9.1e-031;    # Rest mass of electron, kg

#Calculations
m = 11*m0;    # Mass of relativistically moving electron, kg
E_k = (m-m0)*c**2/(1.6e-019*1e+06);    # Kinetic energy of moving electron, MeV
# As m = m0/sqrt(1-v^2/c^2), solving for v
v = c*sqrt(1-(m0/m)**2);    # The velocity of the moving electron, m/s
p = m*v;    # Momentum of moving electron, kg-m/s

#Results
print "The kinetic energy of moving electron = %4.2f MeV"%E_k
print "The momentum of moving electron = %4.2e kg-m/s"%p

The kinetic energy of moving electron = 5.12 MeV
The momentum of moving electron = 2.99e-21 kg-m/s


## Example 1.13, Page 32¶

In [20]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, m/s
E0 = 0.5;    # Rest energy of the electron, MeV

#Calculations
v1 = 0.6*c;    # Initial velocity of the electron, m/s
v2 = 0.8*c;    # Final velocity of the electron, m/s
W = (1/sqrt(1-v2**2/c**2)-1/sqrt(1-v1**2/c**2))*E0;    # The amount of work to be done to increase the speed of the electron, MeV

#Result
print "The amount of work to be done to increase the speed of an electron = %4.2e J"%(W*1e+06*1.6e-019)

The amount of work to be done to increase the speed of an electron = 3.33e-14 J


## Example 1.14, Page 33¶

In [15]:
from math import *

#Variable declaration
c = 1;    # Assume speed of light in vacuum to be unity, unit
m0 = 1;    # For simplicity assume rest mass of the particle to be unity, unit

#Calculations
v = c/sqrt(2);    # Given speed of the particle, m/s
gama = 1/sqrt(1-v**2/c**2);    # Relativistic factor
m = gama*m0;   # The relativistic mass of the particle, unit
p = m*v;    # The relativistic momentum of the particle, unit
E = m*c**2;    # The relativistic total eneryg of the particle, unit
E_k = (m-m0)*c**2;    # The relativistic kinetic energy of the particle, unit

#Results
print "The relativistic mass of the particle = %5.3fm"%m
print "The relativistic momentum of the particle = %1.0gm0c"%p
print "The relativistic total energy of the particle = %5.3fm0c^2"%E
print "The relativistic kinetic energy of the particle = %5.3fm0c^2"%E_k


The relativistic mass of the particle = 1.414m
The relativistic momentum of the particle = 1m0c
The relativistic total energy of the particle = 1.414m0c^2
The relativistic kinetic energy of the particle = 0.414m0c^2


## Example 1.15, Page 34¶

In [23]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, unit
m0 = 9.1e-031;    # Rest mass of the electron, kg
m = 1.67e-027;    # Rest mass of the proton, kg

#Calculations
# As m = m0/sqrt(1-v^2/c^2), solving for v
v = c*sqrt(1-(m0/m)**2);    # Velocity of the electron, m/s

#Result
print "The velocity of the electron to have its mass equal to mass of the proton = %5.3e m/s"%v

The velocity of the electron to have its mass equal to mass of the proton = 3.000e+08 m/s


## Example 1.17, Page 35¶

In [24]:
from math import *

#Variable declaration
c = 3e+008;    # Speed of light in vacuum, unit
m0 = 9.1e-031;    # Rest mass of the electron, kg
E_k = 0.1*1e+006*1.6e-019;    # Kinetic energy of the electron, J

#Calculations&Results
v = sqrt(2*E_k/m0);    # Classical speed of the electron, m/s
print "The classical speed of the electron = %5.3e m/s"%v
# As E_k = (m-m0)*c^2 = (1/sqrt(1-v^2/c^2)-1)*m0*c^2, solving for v
v = c*sqrt(1-(m0*c**2/(E_k+m0*c**2))**2);    # Relativistic speed of the electron, m/s
print "The relativistic speed of the electron = %5.3e m/s"%v

The classical speed of the electron = 1.875e+08 m/s
The relativistic speed of the electron = 1.644e+08 m/s