Chapter 2 Solutions Modern Physics 4th Edition

u²/c² = u²₁(1-v²/c²)\nu²/c² = v²\n√(1-u²² + v²/c²)\n = √(1- v²/c² - u²₂/c²)(1-v²/c²)\n = (1-v²/c²)(1-u²₁/c²)\nP₂₁ = \\frac{m u₁}{√{1-v²/c² + u²₁/c²}}\n = - \\frac{m u₁}{√{1-v²/c²√{1-u²₀/c²}}\n = -m u₁√{1-u²₀/c²} = -p_{da}\n\nd(γmu) = m(αx dy + γdu)\n = -m[(-\\frac{1}{2})(\\frac{-2λ}{c²})^{-3/2}(1-\\frac{u²}{c²})] + \n = m[\\frac{x²}{c²}(1-\\frac{u²}{c²})^{-1/2}]\n\ny = \\frac{1}{√{1-u²/c²}}\n\\frac{1}{√{1-(0.6c)²/c²}} = 1.25(0.511 MeV)\n= 0.639 MeV\nE_k = γmc² = 1.25(0.511 MeV) = 0.128 MeV\nE_k = (γ - 1)mc² = 0.25(0.511 MeV) • 0.128 MeV E_k = (γ-1)mc² = [(1-u²/c²)^{-γ}-1]mc²\nE_k = [1-(0.5)²]^{-γ}-1]mc² = 0.155mc²\nE_k = [1-(0.9)²]^{-γ}-1]mc² = 1.29mc²\nE_k = [1-(0.99)²]^{-γ}-1]mc² = 6.09mc²\nΔE = Δmc² … Δm = ΔE/c²\n10J / (3.0×10⁸ m/s)² = 1.1×10^{-16} kg\nm(u) = m/√{1-u²/c²}\nm(u) = 5m\n\\frac{m}{√{1-u²/c²}} = 5m → 1-\\frac{u²/c²}{1/25}\nu²/c² = 1-1/25 = 0.960 → u/c = 0.980\nm(u) = 20m\n\\frac{m}{√{1-u²/c²}} = 20m → 1-\\frac{u²/c²}{1/400}\nu²/c² = 1-1/400 = 0.9975 → u/c = 0.99870\nv = /Earth - moon distance/√{time} = 3.8×10⁸ m/1.5 s = 0.84c\nmc²(proton) = 938.3 MeV γ = √{1-0.84²} = 1.87 E_k = 938. MeV(1.87-1) • 813 MeV\nm(u) = m\n 938.3 MeV/c^2 = 1.730×10^3 MeV/c^2 = 1.730 GeV/c^2\n√(1-u²/c²)\nE_k = \\frac{1}{2} mv² = \\frac{1}{2} 938. MeV/c²(0.84c)² = 331 MeV\n813 MeV - 331 MeV \\times 100 = 59%\n\nE_k = mc²(γ-1)\nE_k(u2) - E_k(u1) = W21 = mc²(γ(u2)-1 - mc²(γ(u1)-1)\nW21 = 938. MeV[1-0.96²]^{-1/2} ·[1-0.15²]^{-1/2} = 1.51 MeV\nW21 = 938. MeV[1-0.86²]^{-1/2} ·[1-0.85²]^{-1/2} = 57.6 MeV\nW21 = 938. MeV[1-0.96²]^{-1/2} ·[1-0.95²]^{-1/2} = 3.35×10³ MeV • 3.35 GeV\nE = γmc²\n200 GeV×197 = 3.94×10⁴ GeV = γ(0.938 GeV) where mc²(proton) = 0.938 GeV\nγ = \\frac{1}{√{1-v²/c²}} = \\frac{3.94×10⁴ GeV}{0.938 GeV} = 4.20×10⁴\n\\frac{v}{c} = 1 - \\frac{1}{2γ²}\n\\frac{v}{c} = 1 - \\frac{1}{2(4.20×10⁴)²} = 1 - 0.00000002834 Thus, v = 0.999999999717 c\nE = pc for E >> mc²\np = \\frac{E}{c} = 3.94×10⁴ GeV/c E = γmc²\nγ = E/mc² = 1400 MeV/938 MeV = 1.4925\n\nγ = 1/√(1-u²/c²) → 1.4925 → 1-u²/c² = 1/(1.4925)²\nu²/c² = 1 - 1/(1.4925)² = 0.551 → u = 0.74c\n\nE² = (pc)² + (mc²)²\np = 1/c[ E² - (mc²)²]^{1/2} = 1/c(1400 MeV)² - 938 MeV)²^{1/2} = 1040 MeV/c\n\nγ = E/mc² = 2.1/√(1-u²/c²) - 1 - u²/c² = 1/4\n\hu²/c² = 1 - 1/4 = 0.75 → u = 0.866c\np = 1/c[E² - (mc²)²]^{1/2}\np = 1/c[2mc²² - (mc²)²]^{1/2}\n\nu = 2.2 x 10⁶ m/s and γ = 1/√(1-u²/c²)\nEk = 0.5110 MeV(γ-1) = 0.5110 MeVγ[1/√(1-u²/c²)-1] = 0.5110(2.689 x 10⁻⁵) = 1.3741 x 10⁻⁵ MeV\n\nEk(classical) = 1/2 mu² = 1/2 mc²u²/c² = 0.5110 MeV/2)2.2 x 10²\n= 1.374 x 10⁻⁵ MeV\n\n% difference = 1 x 10⁹ / 1.3741 x 10⁻⁵ = 0.0073% p(classical) = mu = mc² אַ (2.2 x 10⁶ /3.0 x 10⁸)\n= 3.74733 x 10⁻⁷\n% difference = 1.2 x 10¹⁰ / 3.74745 x 10⁻⁷\n\n60 W = 60/1.3 x 10⁷ s\nm = E/c² = 1.896 x 10⁹J = 2.1 x 10⁻⁸ kg = 2.1 x 10⁻⁵ g = 21 µg\n\n⁴He → ³H + p + e\nQ = m₁ᵗH + mₒʲ + mₗ⁴He\n= 2809.450 MeV + 938.280 MeV + 0.511 MeV - 3728.424 MeV = 19.817 MeV\n\n³H → ²H + n\n= 32.014102 + 2.17 - 1.008 = 0.006718 x 931.5 MeV/u = 6.26 MeV Δm = m(^4He) - 2m(^3H) = m(^4He)c^2 - 2m(^3H)c^2 / c^2\n= {√(3727.409 MeV) - 2 × 1875.628 MeV/931.5 MeV}.\n= -0.0256 u\nΔE = |Δm|c^2 = (0.0256 u)(931.5 MeV/u c^2) = 23.8 MeV\ndN / dt = P / ΔE = 1 W / ΔE = 2.62 x 10^{11} / s\ndm / dt = 1 / c^2 dE = 100 x 10^6 W / (9 x 10^6 J/kg) = 4.0 x 10^{-6} kg/h / h\nE^1 = E^1,\n(ρc)c^2 + 2m_pc^2 + m_c^2 = (ρc)^2 + 2m_pc^2 + m_c^2\n0 = 2m_pc^2 × 2m_pc^2 + m_c^2\n0 = m_c^2 + m_p/2m_p\nE_i = m_i c^2 + 2 + m_c^2 / 2m_p = 135 MeV(2 + 135/2(938)) = 280 MeV\nΔE = Δmc^2 • Δm = ΔE/c^2 • 3.57 x 10^{-35} g\n(200 x 10^6 eV)/(5.61 x 10^{32} eV/g). E_K = 497.7 MeV - 2(139.6 MeV) = 218.5 MeV\n218.5 MeV/2 = 109.25 MeV\n1.0 W • 1.0 J/s → p = E/c = (1.0 J/s)/c\nΔp = (1.0 J/s) / c\nFΔt = Δp where Δt = 1 s → F = (1.0 J/s) / (c Δm) = (1.0/c)N = 3.3 x 10^{-9}N\nm = F/g = 3.3 x 10^{-9}N / (9.8 m/s^2) = 3.4 x 10^{-10} kg = 0.34 μg\nF = 6.6 x 10^{-9}, m = 0.68 μg\n(2mc^2)^2 = E_i^2 + (ρc)^2\np_i = 0, E_i = 2mc = 2(0.511 MeV) = 1.022 MeV\n(2mc^2)^2 = E_f^2 + (p_f c)^2\nE^2 = (pc)^2 + (mc)^2\nE = |(pc)^2 + (mc)^2|^{1/2} + (pc/mc^2)^{1/2} = mc^2|1 + (p^2/mc^2) + ...\n= mc^2|1 + p^2/2m| = mc^2 + p^2/2m. E^2 = (pc)^2 + mc^2\n(pc)^2 + E^2 - (mc^2)^2 = (5 MeV)^2 - (0.511 MeV)^2 = 24.74\nOr p = √(24.74/c) = 4.97 MeV/c\nE = γmc^2 + γ = E/mc^2 = 1/√(1 - u^2/c^2) → 1 - u^2/c^2 = mc^2/E^2\nu/c = 1 - (mc^2/E^2)^{1/2} = 1 - (0.511/5.0)^{1/2} = 0.995\nE^2 = (pc)^2 + mc^2\n(1746 MeV)^2 - (500 MeV)^2 + (mc^2)^2\nmc^2 = (1746 MeV)^2 - (500 MeV)^2}^{1/2} = 1673 MeV → m = 1673 MeV/c^2\nE = γmc^2 + γ = 1/√(1 - u^2/c^2) = E/mc^2\nu/c = 1 - (mc^2/E^2)^{1/2} = 1 - (1673 MeV/1746 MeV)^{1/2} = 0.286 → u = 0.286c\nBqR = myu = p\nB = myu/qR and E = γmc^2\nu/c = 1 - (mc^2/E^2)^{1/2} = 1 - (0.511 MeV/4.0 MeV)^{1/2} = 0.992\nγ = 1/√(1 - u^2/c^2) = 1/√(1 - (0.992)^2) = 7.83\nB = (9.11 x 10^{-31} kg)(7.83)(0.992c) / (1.60 x 10^{-19} C)(4.2 x 10^{-2} m)\nγm = 7.83. p = qBR = e(0.5T)(2.0) \\times \\frac{3.0 \\times 10^{8} m/s}{c} = 300 MeV/c\nE_{k} = E - mc^{2} = \\sqrt{(pc)^{2} + (mc^{2})^{2}} - mc^{2}\n = \\sqrt{(300 MeV)^{2} + (938.28 MeV)^{2}} - 938.28 MeV\\n = 46.8 MeV\n\\alpha = 2GM/c^{2} R\n = \\frac{6.37 \\times 10^{6} m}{5.98 \\times 10^{24} kg} \\n\\alpha = \\frac{2(6.67 \\times 10^{-11} N m^{2}/kg^{2})(5.98 \\times 10^{24} kg)} {(3.00 \\times 10^{8} ms^{-1})^{2}(6.37 \\times 10^{6} m)} = 1.39 \\times 10^{-9} radians\n\\Delta \\phi = \\frac{6 \\pi GM}{c^{2} R} = 6 \\pi \\cdot 6.67 \\times 10^{-11} N m^{2}/kg^{2}(1.99 \\times 10^{30} kg)\n\\frac{(3.00 \\times 10^{8} m/s)^{2}}{(6.37 \\times 10^{6} m)} = 3.64 \\times 10^{-8} radians/century = 7.55 \\times 10^{-3} arc seconds/century\nf_{0} - f = f_{0} gh/c^{2}\n = \\frac{GM}{R^{2}} - \\frac{9.9 m/s^{2}}{6.623} = 0.223 m/s^{2}\n\\h = 6.623R_{e} = 6.623 \\times 10^{6} m = 4.22 \\times 10^{7} m\n\\Delta f - f = 9.375 \\times 10^{10} Hz(0.223 m/s^{2})(4.22 \\times 10^{7}/(3.00 \\times 10^{8})^{2} = 0.980 Hz\nf = f_{0} - f = 9.374999999 \\times 10^{9} Hz \\alpha = 2Gm/c^{2} R = \\frac{2(6.67 \\times 10^{-11} N \\cdot m^{2}/kg^{2})(3)(1.99 \\times 10^{30} kg)}{(3.00 \\times 10^{8} m/s)^{2}(10^{7} m)}\n = 8.85 \\times 10^{-4} radians = 0.051°\\n2\\alpha = 0.102°\nv = \\frac{2\\pi R}{T} = 2 \\pi \\left(6.37 \\times 10^{6} m\\right)/(90 min \\times 60 s/min) = 7.42 \\times 10^{3} m/s\n\\Delta r^{'} - \\Delta r = \\Delta r (1 - 1/\\gamma) = \\Delta r(v^{2}/2c^{2})\n\\Delta r - \\Delta r^{'} = \\frac{3.16 \\times 10^{7} s}{(2)(3.0 \\times 10^{8})^{2}} = 0.00965 s = 9.65 ms\n\\Delta f/f_{0} = \\frac{gh}{c^{2}} = \\frac{9.8 m/s^{2}}{3.0 \\times 10^{5} m} = \\frac{3.27 \\times 10^{-11} s/s}{(3.0 \\times 10^{8} m/s)^{2}} = 1.03 ms\n(mc^{2})^{2} = E^{'}^{2} - p^{'}c^{2} \\quad p^{'} = 0\nE^{'} = mc^{2} = 4.6 kg \\cdot (3.0 \\times 10^{8} m/s)^{2}/4 = 4.14 \\times 10^{14} J E = \\gamma m c^{2} \\rightarrow \\gamma = E/m c^{2} = 50 \\times 10^{3} MeV/0.511 MeV = 9.78 \\times 10^{4}\nL = L_{0}/\\gamma = 10^{-2} m\nL_{0} = 9.78 \\times 10^{4} (10^{-2} m) = 978 m\nL_{0} \\cdot \\gamma(978) = 978 \\times 10^{4}(978) = 9.57 \\times 10^{7} m\nL = 1 \\cdot 10^{-2} m / \\gamma = 10^{-2} m / 9.78 \\times 10^{4} = 1.02 \\times 10^{-7} m\nE_{k} = \\gamma m c^{2} - m c^{2}(\\gamma - 1) \\quad E_{k} = mc^{2} = 938 MeV\n\\frac{(mc^{2})^{2}}{E^{2}- (pc)^{2}} \\quad E = \\gamma m c^{2} = 2(938 MeV)\n\\frac{(pc)^{2}}{E^{2} - m^{2}c^{4}} = (2 \\times 938)^{2} - (938)^{2} = 2.46 \\times 10^{5}\n p = (2.64 \\times 10^{5})^{1/2}/c = 1.62 \\times 10^{3} MeV/c\np = \\gamma mu = \\frac{p}{\\gamma m} = 1.62 \\times 10^{3} MeV/c/(2)938 MeV/c^{2} = 0.866 c\np = \\gamma mu \\sqrt{1-u^{2}/c^{2}} = 10^{3} kg/c(1/2)\\sqrt{1 - 0.5^{2}} = 1.73 \\times 10^{11} kg/m/s\np_c = m_{0} u \\sqrt{1 - u^{2}/c^{2}} = 1.73 \\times 10^{11} kg/m/s^{2}\n m_{0} = m_{u /c^{2}} = 1.73 \\times 10^{11} kg/m/s^{2}\n m^{2}c^{2} = \\left(1 - \\frac{u^{2}}{c^{2}} \\gamma^{2}\\right) 1.73 \\times 10^{11} kg/m/s^{2}\n = 1.73 \\times 10^{5} kg^{2}_{u}\\n(10^{6} kg)^{2} + (3.33 \\times 10^{5} kg u')^{2} = 1.73 \\times 10^{11} kg/m/s^{2}\n\\ u' = 1.73 \\times 10^{11} kg/m/s/10^{6} kg = 1.73 \\times 10^{5} m/s = 5.77 \\times 10^{4} c mu = mc/2 = 10^3c/2,\n m_x \nu_x\n u_f = 10^3(c/2)/m_f = 10^3 kg 3.0 x 10^8 m/s / 2.10^6 kg\n E_i = m_f c^2\n E_f = energy of fuel + energy of ship = mc^2 / sqrt{1 - u^2/c^2} + m_f c^2 / sqrt{1 - u_f^2/c^2}\n u = 0.5c and u_f << c, so\n E_f = 1.155 mc^2 + m_f - m_i c^2 = (1.155 - 1)mc^2 + m_f c^2\n ΔE = E_f - E_i = (0.155)103 kg c^2 + 106 kg c^2 - [106 kg c^2]\n ΔE = 155 kg c^2 or 155 kg • ΔE / c^2\n p = 300 BR q/e\n p = 300, 1.5T, 6.37 x 10^6 (1) = 2.87 x 10^9 MeV/c\n T = 2πR/c = 2π(6.37 x 10^6 m_i/c = 0.133 s\n f / f_0 = 1 - GM / c^2 R\n f_o - f = GM / c^2 R = 7 x 10^-4 R = GM/c^2 (7 x 10^-4) = 6.67 x 10^-11 N m^2/kg^2 (2 x 10^30 kg)\n = 2.12 x 10^-6 m\n (3.00 x 10^8 m/s)^2 (7 x 10^-4)\n ρ = M / V = 2 x 10^30 kg / (4π(2.12 x 10^6 m^3 / 3)\n ρ = 5.0 x 10^10 kg/m^3\n E_p = 1.022 MeV\n ρ = E/c = 1.022 MeV/c [1 - (u'/c)^2]^{1/2} = 1 - u^2/c^2 / (1 + u^2/c^2)\n p' = mu'/sqrt{1 - (u'/c)^2} where u' and sqrt{1 - (u'/c)^2}\n p_i' = m 2u/(1 + u^2/c^2) = 2mu/(1 - u^2/c^2)\n p_f' = Mu/(1 - u^2/c^2)^{1/2} = 2mu/(1 - u^2/c^2)^{1/2} Or M = 2m/(1 - u^2/c^2)^{1/2}\n E_i = 2mc^2 / sqrt{1 - u^2/c^2} and E_f = mc^2\n M = 2m/(1 - u^2/c^2)^{1/2} E_i = E_f\n E' = 2mc^2 / (1 + u^2/c^2)\n E_{kin} = m_pc^2(γ - 1),\n u = 0.866\n u' = (u - v)/(1 - u_x v/c^2)\n u' = -2u /(1 + u^2/c^2)\n = -0.990c p_i = 0 = E/c - Mv or v = E/Mc\n\\Delta x = v \\Delta t\n\\Delta r = L/c\n\\Delta x = (E/Mc)(L/c) = EL/Mc^2\n\nx_{CM} = \\frac{M(0) + m(L/2)}{M + m} = \\frac{mL}{2(M + m)}\nm = E/\\left(c^2\\sqrt{1 - E/Mc^2}\\right)\nE << Mc^2, then m = E/c^2\n\nv = E_u^2 = (p_u c)^2 + m_e^2 c^4 = E^2 = (p_e c)^2 + 0\nE_u + E_y = 139.56755 MeV - 105.65839 MeV\nm_h^2 c^2 \\cdot (\\gamma - 1) + E_y = 33.90916 MeV\np_c = E_u^2 + m_e^2 c^2 = 33.90916 - m_e^2 c^2 (\\gamma - 1)\n\\gamma - 1 = \\frac{(33.90916)^2}{2(m_f^2c^2) + 2(33.90916)m_e^2 c^2}\n\n\\gamma - 1 = 0.030\\gamma - 1 = 1.039\nE_u = 4.12 MeV and p_u = \\frac{1}{c} \\left[109.78\\left(2{\\text{MeV}}\\right)\\right]^{1/2} = 29.8 / MeV/c\n\nE_u = 29.8 MeV and p_u = 29.8 MeV/c and\nv_u \\text{ mass } = 250 keV, then E^2 = (p_u c)^2\nE_u + E_u = 139.56755 MeV - 105.65839 MeV - 0.250 MeV = 33.67916 MeV\n\nE_u = 109.78 MeV, p_u = 29.8 MeV/c, E_v = 29.8 MeV, p_v = 29.8 MeV/c F/f_0 = 1 - \\frac{Gm/c^2R}{c = f_1 \\lambda_0}\nc = f \\lambda_0 {\text{ and }} c = f_0\\lambda_0,\n\\frac{c}{\\lambda} \\cdot \\frac{\\lambda_0}{\\lambda_0} = 1 - \\frac{GM/c^2R} = 1 - \\left(6.67 \\times 10^{-11} N \\cdot m^2/kg^2 \\cdot 1.99 \\times 10^{30} kg\\right)\\cdot\\left(\\frac{3.00 \\times 10^8 m/s^2}{6.96 \\times 10^6 m}\\right)\n= 1 - 0.000212 = 0.999788\n\\lambda = \\frac{\\lambda_0}{0.999788} = \\frac{720.00 nm}{0.999788} = 720.15 nm\n\\Delta \\lambda = \\lambda - \\lambda_0 = 0.15 nm\n\nu_y' = \\left(\\frac{u'_y}{\\gamma} (1 - \\frac{v_u}{c^2})^{-1}\\right)\n\na'_y = \\frac{a_y}{\\gamma^2(1 - \\frac{v_u}{c^2})}\n\\frac{da_y}{dt} = \\frac{d(mv_y)}{dt}\nF_x = \\frac{dp_x}{dt} = \\frac{d(mv_y)}{dt} = F'_x = ma'_z because u'_z = 0\nF_x = \\gamma m_a \\cdot \\frac{d\\lambda}{dt} + mv_y \\cdot dt \\cdot (1 - \\frac{v^2}{c^2})^{-1}/dt\nF_x = m_a \\cdot a_x \\cdot \\frac{m}{(1 - \\frac{v^2}{c^2})^{3/2}}\nF_x = \\gamma^3 m_a x u_z' = 0\nF_x = \\gamma^3 m_a z a_x' = F'_x\nF_y = -\\frac{dP_y}{dt} = -\\frac{dY_y v_y}{dt}\nF'_y = m_a' y because u'_y = u'_y = 0\nF_y = \\gamma m_a y\nu'_y = u'_y - 0, a_y = a_y' / \\gamma^2\nF_y = \\gamma m_a y = \\gamma a_y' / \\gamma^2 = m_a' a_y' = F'_y / \\gamma\n\n(Mc^2)^2 = E^2 - (pc)^2 = (Mc^2)^2 + 0\n(2mc^2)^2 = (Mc^2)^2 + 0\n(mc^2)^2 = (Mc^2/2)^2 - p^2c^2 = (Mc^2/2)^2 - (\\gamma u/c)^2\n1 = \\left(\\frac{Mc^2}{2mc^2}\\right)^2 - \\frac{\\gamma u/c^2} = \\gamma^2 = \\left(\\frac{Mc^2}{2mc^2}\\right)^2 = \\frac{1}{1 - u^2/c^2}\n\nu = \\left[1 - \\left(\\frac{2mc^2}{Mc^2}\\right)\\right]^{1/2} c\n\n(Mc^2)^2 = (4mc^2)^2 - (pc)^2\n(2mc^2)^2 = (4mc^2)^2 - (pc)^2\n(pc)^2 = 4mc^2 - (Mc^2)^2 u\nc = pc E * ( 4mc2 2 - ( Mc2 2 ) 1/2 )\n 4mc2\n\n( u ) 2 = ( 4mc2 2 - ( Mc2 )2 ) \n 4mc2 2 - 1 - ( Mc2 4mc2 )2\n u = [ 1 - ( Mc2 4mc2 )2 ] 1/2 c\n -\n -