{
"metadata": {
"name": "Chapter2",
"signature": "sha256:be87f6a340484dd1a4e5b8f9343e232694681e2bed2590e22d8288691c17dddf"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 2:The Special Theory of Relativity"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.1, Page 22"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of varible\n",
"v1=60.0; v2=40.0 #Velocities of cars wrt to observer in km/hr\n",
"\n",
"#calculation\n",
"vr=v1-v2; #relative velocity\n",
"\n",
"#result\n",
"print\"The value of relative velocity in km/h. is\",round(vr,3);\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The value of relative velocity in km/h. is 20.0\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.2, Page 22"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import atan, pi\n",
"import numpy as np\n",
"Va_w=[320.0,0.0]; Vw_g=[0.0, 65.0]; #Vp/q=[X Y]=>velocity of object p wrt q along X(east) and Y(north) directions.\n",
"\n",
"#calculation\n",
"Va_g=Va_w + Vw_g; #net velocity\n",
"k=np.linalg.norm(Va_g) #magnitude\n",
"s=atan(Va_g[3]/Va_g[0])*180.0/pi; #angle in rad*180/pi for conversion to degrees\n",
"\n",
"#result\n",
"print \"the velocity in x direction in Km/h is\", Va_w[0],\"in y direction in km/h is\",Vw_g[1]\n",
"print\"The magnitude of velocity Va/g(airplane wrt ground) in Km/h is\",round(k,3),\" at \",round(s,3),\" degrees north of east.\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The magnitude of velocity Va/g(airplane wrt ground) in Km/h is 326.535 at 11.482 degrees north of east.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.4, Page 28"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"Lo=100.0*(10**3);c=3.0*(10**8); #Given values//all the quantities are converted to SI units \n",
"d=2.2*(10**-6); #time between its birth and decay\n",
"\n",
"#calculation\n",
"t=Lo/c #where Lo is the distance from top of atmosphere to the Earth. c is the velocity of light. t is the time taken\n",
"u=sqrt(1-((d/t)**2)); # using time dilation fromula for finding u where u is the minimum velocity in terms of c;\n",
"\n",
"#result\n",
"print\"Hence the minimum speed required in c is\",round(u,6);\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Hence the minimum speed required in c is 0.999978\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.5, Page 30"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#intiation of variable\n",
"from math import sqrt\n",
"Lo=100.0*(10**3); #Lo is converted to Km\n",
"u=0.999978; #//u/c is taken as u since u is represented in terms of c. \n",
"\n",
"#calculation\n",
"L=Lo*(sqrt(1-u**2)); # from the length contraction formula\n",
"\n",
"#result\n",
"print\"Hence the apparent thickness of the Earth's surface in metres. is\",round(L,3)\n",
"print\"answer is slightly different in the book\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Hence the apparent thickness of the Earth's surface in metres. is 663.321\n",
"answer is slightly different in the book\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.6, Page 32"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"L=65.0; c=3*10**8;u=0.8*c; \n",
"\n",
"#calculation\n",
"t=L/u ; #The value of time taken as measured by the observer\n",
"\n",
"#result\n",
"print\"The time for rocket to pass a point as measured by O in musec is \",round(t*10**6,3); #The value of time taken as measured by the observer\n",
"\n",
"#partb\n",
"Do=65.0; #given length\n",
"Lo= L/sqrt(1-(u/c)**2); #contracted length of rocket\n",
"\n",
"#result\n",
"print\"Actual length according to O is \",round(Lo,3);\n",
"\n",
"#partc\n",
"D=Do*(sqrt(1-(u/c)**2)); #contracted length of platform.\n",
"\n",
"#result\n",
"print\"Contracted length according to O'' is\",round(D,3);\n",
"\n",
"#partd\n",
"t1=Lo/u; #time needed to pass according to O'.\n",
"print \"Time taken according to O is \",t1\n",
"\n",
"#part 3\n",
"t2=(Lo-D)/u; #time intervals between the two instances\n",
"print\"Time taken according to O'' is \",t2;\n",
"print'The value of t1 and t2 does not match with textbook exactly';"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time for rocket to pass a point as measured by O in musec is 0.271\n",
"Actual length according to O is 108.333\n",
"Contracted length according to O'' is 39.0\n",
"Time taken according to O is 4.51388888889e-07\n",
"Time taken according to O'' is 2.88888888889e-07\n",
"The value of t1 and t2 did not match\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.7, Page 35"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"v1=0.6; u=0.8; c=1.0; # all the values are measured in terms of c hence c=1\n",
"\n",
"#calculation\n",
"v= (v1+u)/(1+(v1*u/c**2));\n",
"\n",
"#result\n",
"print \"The speed of missile as measured by an observer on earth in c is\",round(v,3);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The speed of missile as measured by an observer on earth in c is 0.946\n"
]
}
],
"prompt_number": 23
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.8, Page 37"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"w1=600.0;w2=434.0; # w1=recorded wavelength;w2=actual wavelength\n",
" # c/w1 = c/w2 *(sqrt(1-u/c)/(1+u/c))\n",
" \n",
"#calculation\n",
"k=w2/w1;\n",
"x=(1-k**2)/(1+k**2); #solving for u/c\n",
"\n",
"#result\n",
"print\"The speed of galaxy wrt earth in c is\",round(x,3);\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The speed of galaxy wrt earth in c is 0.313\n"
]
}
],
"prompt_number": 24
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.9, Page 39"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"v1x=0.6;v1y=0.0;v2x=0.0;v2y=.8;c=1.0; # all the velocities are taken wrt c\n",
"v21x=(v2x-v1x)/(1-(v1x*v2x/c**2)); #using lorentz velocity transformation\n",
"v21y=(v2y*(sqrt(1-(v1x*c)**2)/c**2))/(1-v1y*v2y/c**2) \n",
"\n",
"#result\n",
"print\"The velocity of rocket 2 wrt rocket 1 along x and y directions is\",round(v21x,3),\" c &\", round(v21y,3),\"c respectively\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The velocity of rocket 2 wrt rocket 1 along x and y directions is -0.6 c & 0.64 c respectively\n"
]
}
],
"prompt_number": 27
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.10, Page 40"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"u=0.8*c;L=65.0;c=3.0*10**8; #all values are in terms of c\n",
"t=u*L/(c**2*(sqrt(1-((u/c)**2)))); #from the equation 2.31 \n",
"\n",
"#result\n",
"print\"The time interval between the events is\",t, \"sec which equals\",round(t*10**6,3),\"musec.\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time interval between the events is 2.88888888889e-07 sec which equals 0.289 musec.\n"
]
}
],
"prompt_number": 30
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.11, Page 41"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"m=1.67*10**-27;c= 3*10**8;v=0.86*c; #all the given values and constants\n",
"\n",
"#calculation\n",
"p=m*v/(sqrt(1-((v/c)**2))); # in terms of Kgm/sec\n",
"\n",
"#result\n",
"print\"The value of momentum was found out to be in Kg-m/sec.\\n\",p;\n",
"\n",
"#part 2\n",
"c=938.0;v=0.86*c;mc2=938.0 # all the energies in MeV where mc2= value of m*c^2\n",
"pc=(mc2*(v/c))/(sqrt(1-((v/c)**2))); #expressing in terms of Mev\n",
"\n",
"#result\n",
"print\"The value of momentum was found out to be in Mev.\",round(pc,3);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The value of momentum was found out to be in Kg-m/sec.\n",
"8.44336739668e-19\n",
"The value of momentum was found out to be in Mev. 1580.814\n"
]
}
],
"prompt_number": 34
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.12, Page 47"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"pc=1580.0; mc2=938.0;E0=938.0; # all the energies in MeV mc2=m*c^2 and pc=p*c\n",
"\n",
"#result\n",
"E=sqrt(pc**2+mc2**2); \n",
"K=E-E0; #value of possible kinetic energy\n",
"\n",
"#result\n",
"print\"The relativistic total energy in MeV. is\",round(E,3); #value of Energy E\n",
"print\"The kinetic energy of the proton in MeV.\",round(K,3);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The relativistic total energy in MeV. is 1837.456\n",
"The kinetic energy of the proton in MeV. 899.456\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.13, Page 47"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"E=10.51; mc2=0.511; #all the values are in MeV\n",
"\n",
"#calculation\n",
"p=sqrt(E**2-mc2**2); #momentum of the electron\n",
"v=sqrt(1-(mc2/E)**2); #velocity in terms of c\n",
"\n",
"#result\n",
"print\"The momentum of electron in MeV/c is\",round(p,3); \n",
"print\"The velocity of electron in c is\",round(v,5);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The momentum of electron in MeV/c is 10.498\n",
"The velocity of electron in c is 0.99882\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.14, Page 47"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"k=50;mc2=0.511*10**-3;c=3.0*10**8; # all the values of energy are in GeV and c is in SI units\n",
"\n",
"#calculation\n",
"v=sqrt(1-(1/(1+(k/mc2))**2)); #speed of the electron in terms of c\n",
"k=c-(v*c); #difference in velocities\n",
"\n",
"#result\n",
"print\"Speed of the electron as a fraction of c*10^-12 is.\",round(v*10**12,3); # v=(v*10^12)*10^-12; so as to obtain desired accuracy in the result\n",
"print\"The difference in velocities in cm/s.\",round(k*10**2,3);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Speed of the electron as a fraction of c*10^-12 is. 9.99999999948e+11\n",
"The difference in velocities in cm/s. 1.567\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.15, Page 48"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt, pi\n",
"r=1.5*10**11; I=1.4*10**3; #radius and intensity of sun\n",
"\n",
"#calculation\n",
"s=4*pi*r**2 #surface area of the sun\n",
"Pr=s*I # Power radiated in J/sec\n",
"c=3.0*10**8; #velocity of light\n",
"m=Pr/c**2 #rate of decrease of mass\n",
"m=round(m,2)\n",
"\n",
"#result\n",
"print\"The rate of decrease in mass of the sun in kg/sec. is %.1e\" %m;"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The rate of decrease in mass of the sun in kg/sec. is 4.4e+09\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.16, Page 48"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import pi, sqrt\n",
"K=325; mkc2=498; #kinetic energy and rest mass energy of kaons\n",
"mpic=140.0; #given value\n",
"\n",
"#calculation\n",
"Ek=K+mkc2; \n",
"pkc=sqrt(Ek**2-mkc2**2); \n",
"#consider the law of conservation of energy which yields Ek=sqrt(p1c^2+mpic^2)+sqrt(p2c^2+mpic^2)\n",
"#The above equations (4th degree,hence no direct methods)can be solved by assuming the value of p2c=0.\n",
"p1c=sqrt(Ek**2-(2*mpic*Ek));\n",
"#consider the law of conservation of momentum. which gives p1c+p2c=pkc implies\n",
"p2c=pkc-p1c;\n",
"k1=(sqrt(p1c**2+(mpic**2))-mpic); #corresponding kinetic energies\n",
"k2=(sqrt((p2c**2)+(mpic**2))-mpic);\n",
"\n",
"#result\n",
"print\"The corresponding kinetic energies of the pions are\", k1,\" MeV and\",round(k2,3),\" MeV.\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The corresponding kinetic energies of the pions are 543.0 MeV and 0.627 MeV.\n"
]
}
],
"prompt_number": 49
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2.17, Page 49"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#initiation of variable\n",
"from math import sqrt\n",
"mpc2=938.0;c=3.0*10**8; #mpc2=mp*c^2,mp=mass of proton\n",
"\n",
"#calculation\n",
"Et=4*mpc2; #final total energy\n",
"E1=Et/2;E2=E1; #applying conservation of momentum and energy\n",
"v2=c*sqrt(1-(mpc2/E1)**2); #lorentz transformation\n",
"u=v2;v=(v2+u)/(1+(u*v2/c**2)); \n",
"E=mpc2/(sqrt(1-(v/c)**2));\n",
"K=E-mpc2;\n",
"\n",
"#result\n",
"print\"The threshold kinetic energy in Gev\",round(K/10**3,3);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The threshold kinetic energy in Gev 5.628\n"
]
}
],
"prompt_number": 50
}
],
"metadata": {}
}
]
}