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witw78
Apr 21st, 2025
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  1. [
  2.     {
  3.         "question_structure": {
  4.             "context": "A light spring is fixed at one end. When a force of magnitude $F_1 = 10\\mathrm{N}$ is applied to compress the other end, the equilibrium length is $l_1 = 9\\mathrm{cm}$. When a force of magnitude $F_2 = 20\\mathrm{N}$ is applied to stretch the spring, the equilibrium length becomes $l_2 = 12\\mathrm{cm}$. The stretching or compression of the spring is within the elastic limit. The spring was slightly rusted, which might affect its elastic coefficient, but it was assumed to be negligible in the experiment.",
  5.             "sub_question_1": "Find the original length $l_0$ of the spring.",
  6.             "sub_question_2": "Determine the spring constant $k$ of the spring."
  7.         },
  8.         "sub_questions": {
  9.             "sub_question_1": "Find the original length $l_0$ of the spring.",
  10.             "sub_question_2": "Determine the spring constant $k$ of the spring."
  11.         },
  12.         "answer": [
  13.             "10 cm",
  14.             "1000 N/m"
  15.         ]
  16.     },
  17.     {
  18.         "question_structure": {
  19.             "context": "An object is released from rest at a height of $45\\mathrm{m}$ above the ground, undergoing free fall motion, with the acceleration due to gravity taken as $\\mathrm{g}\\!=\\!10\\mathrm{m}/\\mathrm{s}^{2}$. There was a slight air resistance during the free - fall motion, although it was considered to have no significant impact on the result.",
  20.             "sub_question_1": "What is the duration of time it takes for the object to reach the ground?",
  21.             "sub_question_2": "What is the magnitude of the object's velocity upon impact with the ground?",
  22.             "sub_question_3": "What is the distance the object falls during its last 1 second of motion?"
  23.         },
  24.         "sub_questions": {
  25.             "sub_question_1": "What is the duration of time it takes for the object to reach the ground?",
  26.             "sub_question_2": "What is the magnitude of the object's velocity upon impact with the ground?",
  27.             "sub_question_3": "What is the distance the object falls during its last 1 second of motion?"
  28.         },
  29.         "answer": [
  30.             "3s",
  31.             "30m/s",
  32.             "25m"
  33.         ]
  34.     },
  35.     {
  36.         "question_structure": {
  37.             "context": "Cars A and B are $600\\mathrm{m}$ apart and are traveling towards each other on a straight road at speeds of $10\\mathrm{m/s}$ and $20\\mathrm{m/s}$, respectively. A bird flies back and forth between the two cars at a speed of $20\\mathrm{m/s}$. The bird starts from the front of Car A, flies towards Car B, and immediately turns around upon reaching the front of Car B to fly back towards Car A. This back-and-forth flight continues until the two cars meet. The road surface where the cars were moving had a small coefficient of friction, but it was ignored in this problem.",
  38.             "sub_question_1": "The distance this little bird flies",
  39.             "sub_question_2": "The average speed of this small bird's flight"
  40.         },
  41.         "sub_questions": {
  42.             "sub_question_1": "The distance this little bird flies",
  43.             "sub_question_2": "The average speed of this small bird's flight"
  44.         },
  45.         "answer": [
  46.             "$400\\mathrm{m}$",
  47.             "$10\\mathrm{m/s}$"
  48.         ]
  49.     },
  50.     {
  51.         "question_structure": {
  52.             "context": "A proton with mass $m_p$ and energy $E_p$ collides head-on with a photon of energy $E_b$. The collision results in the formation of a new particle with mass $m_\\Delta$. This is a one-dimensional collision that conserves both relativistic energy and relativistic momentum. You may use the following constants where needed:\n$g=9.8 \\mathrm{~N} / \\mathrm{kg}$\n$k=1 / 4 \\pi \\epsilon_{0}=8.99 \\times 10^{9} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{C}^{2}$\n$c=3.00 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$\n$N_{\\mathrm{A}}=6.02 \\times 10^{23}(\\mathrm{~mol})^{-1}$\n$\\sigma=5.67 \\times 10^{-8} \\mathrm{~J} /\\left(\\mathrm{s} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{K}^{4}\\right)$\n$1 \\mathrm{eV}=1.602 \\times 10^{-19} \\mathrm{~J}$\n$m_{e}=9.109 \\times 10^{-31} \\mathrm{~kg}=0.511 \\mathrm{MeV} / \\mathrm{c}^{2}$\n$m_{p}=1.673 \\times 10^{-27} \\mathrm{~kg}=938 \\mathrm{MeV} / \\mathrm{c}^{2}$\n$\\sin \\theta \\approx \\theta-\\frac{1}{6} \\theta^{3}$ for $|\\theta| \\ll 1$\n$G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$\n$k_{\\mathrm{m}}=\\mu_{0} / 4 \\pi=10^{-7} \\mathrm{~T} \\cdot \\mathrm{m} / \\mathrm{A}$\n$k_{\\mathrm{B}}=1.38 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$\n$R=N_{\\mathrm{A}} k_{\\mathrm{B}}=8.31 \\mathrm{~J} /(\\mathrm{mol} \\cdot \\mathrm{K})$\n$e=1.602 \\times 10^{-19} \\mathrm{C}$\n$h=6.63 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}=4.14 \\times 10^{-15} \\mathrm{eV} \\cdot \\mathrm{s}$\n$(1+x)^{n} \\approx 1+n x$ for $|x| \\ll 1$\n$\\ln (1+x) \\approx x$ for $|x| \\ll 1$\n$\\cos \\theta \\approx 1-\\frac{1}{2} \\theta^{2}$ for $|\\theta| \\ll 1$\nvelocity parameter $\\beta = \\frac{v}{c}$\nLorentz factor $\\gamma = \\frac{1}{\\sqrt{1-\\beta^{2}}}$\nrelativistic momentum $p = \\gamma \\beta m c$\nrelativistic energy $E = \\gamma m c^{2}$\nrelativistic doppler shift $\\frac{f}{f_{0}} = \\sqrt{\\frac{1-\\beta}{1+\\beta}}$\nRelativistic particles obey the mass-energy relation: $E^{2}=(p c)^{2}+\\left(m c^{2}\\right)^{2}$ During the particle collision, there was a very small amount of energy loss due to a brief magnetic field interference, which was not accounted for in the calculations.",
  53.             "sub_question_1": "Determine $E_p$ in terms of $m_p$, $m_{\\Delta}$, and $E_b$. You may assume that $E_b$ is small.",
  54.             "sub_question_2": "In this case, the photon energy $E_b$ is that of the cosmic background radiation, which is an EM wave with wavelength $1.06 \\mathrm{~mm}$. Determine the energy of the photons, writing your answer in electron volts.",
  55.             "sub_question_3": "Assuming this value for $E_b$, what is the energy of the proton, in electron volts, that will allow the above reaction? This sets an upper limit on the energy of cosmic rays. The mass of the proton is given by $m_p c^{2}=938 \\mathrm{MeV}$ and the mass of the $\\Delta$ is given by $m_{\\Delta} c^{2}=1232 \\mathrm{MeV}$."
  56.         },
  57.         "sub_questions": {
  58.             "sub_question_1": "Determine $E_p$ in terms of $m_p$, $m_{\\Delta}$, and $E_b$. You may assume that $E_b$ is small.",
  59.             "sub_question_2": "In this case, the photon energy $E_b$ is that of the cosmic background radiation, which is an EM wave with wavelength $1.06 \\mathrm{~mm}$. Determine the energy of the photons, writing your answer in electron volts.",
  60.             "sub_question_3": "Assuming this value for $E_b$, what is the energy of the proton, in electron volts, that will allow the above reaction? This sets an upper limit on the energy of cosmic rays. The mass of the proton is given by $m_p c^{2}=938 \\mathrm{MeV}$ and the mass of the $\\Delta$ is given by $m_{\\Delta} c^{2}=1232 \\mathrm{MeV}$."
  61.         },
  62.         "answer": [
  63.             "$\\frac{m_{\\Delta}^{2}-m_{p}^{2}}{4 E_{b}}$",
  64.             "$\\approx 1.17 \\times 10^{-3} \\mathrm{eV}$",
  65.             "$E_{p} \\approx 1.4 \\times 10^{20} \\mathrm{eV}$"
  66.         ]
  67.     },
  68.     {
  69.         "question_structure": {
  70.             "context": "China's first lunar exploration satellite, \"Chang'e-1,\" was launched in 2007 from the Xichang Satellite Launch Center by the \"Long March 3A\" carrier rocket. Assuming the satellite's orbit is circular and its altitude above the lunar surface is equal to the Moon's radius, it is known that the satellite's orbital period around the Moon is T, the Moon's radius is R, and the gravitational constant is G. The lunar surface has a thin layer of dust, which might cause a small change in the gravitational field, but it was not considered in this model.",
  71.             "sub_question_1": "What is the mass of the Moon?",
  72.             "sub_question_2": "What is the density of the Moon?"
  73.         },
  74.         "sub_questions": {
  75.             "sub_question_1": "What is the mass of the Moon?",
  76.             "sub_question_2": "What is the density of the Moon?"
  77.         },
  78.         "answer": [
  79.             "$\\frac{32\\pi^{2}R^{3}}{G T^{2}}$",
  80.             "$\\frac{24\\pi}{G T^{2}}$"
  81.         ]
  82.     },
  83.     {
  84.         "question_structure": {
  85.             "context": "A thin string of length L is fixed at its upper end, and a small ball of mass m and charge q is attached to the lower end. The ball is placed in a uniform electric field pointing horizontally to the right. Initially, the string and the ball are held horizontally, and then released. The ball swings downward from rest. When the string has rotated through an angle of 60°, the velocity of the ball at point B is exactly zero. There was a weak magnetic field around the charged ball, which could potentially affect its motion, but was assumed to be insignificant.",
  86.             "sub_question_1": "What is the magnitude of the uniform electric field strength?",
  87.             "sub_question_2": "What is the potential difference $U_{AB}$ between points A and B?",
  88.             "sub_question_3": "If the potential at point O is taken as zero, what is the maximum potential energy of the charged ball?"
  89.         },
  90.         "sub_questions": {
  91.             "sub_question_1": "What is the magnitude of the uniform electric field strength?",
  92.             "sub_question_2": "What is the potential difference $U_{AB}$ between points A and B?",
  93.             "sub_question_3": "If the potential at point O is taken as zero, what is the maximum potential energy of the charged ball?"
  94.         },
  95.         "answer": [
  96.             "$\\frac{\\sqrt{3}mg}{q}$",
  97.             "$-\\frac{\\sqrt{3}mgL}{2q}$",
  98.             "$\\frac{\\sqrt{3}mgL}{2}$"
  99.         ]
  100.     },
  101.     {
  102.         "question_structure": {
  103.             "context": "During the Spring Festival holiday, national highways are toll-free, and cars can pass through toll stations without stopping. However, cars are required to maintain a speed no greater than $v_{0} = 6 \\, \\mathrm{m/s}$ when passing through the $x_{0} = 9 \\, \\mathbf{m}$ interval in front of the toll station window. Currently, two cars, Car A and Car B, are traveling at constant speeds of $v_{A} = 20 \\, \\mathrm{m/s}$ and $v_{B} = 34 \\, \\mathrm{m/s}$ respectively on the straight road in front of the toll station. Car A is ahead of Car B. The driver of Car A notices the toll station ahead and begins to decelerate uniformly with an acceleration of $a_{A} = 2 \\, \\mathbf{m/s}^{2}$. The toll station area had a very slight slope, but it was assumed to be a flat road in the analysis.",
  104.             "sub_question_1": "The driver of car A needs to start braking at least how far away from the toll booth window to avoid violating regulations.",
  105.             "sub_question_2": "If the driver of car A brakes and reaches a speed of exactly $6\\mathrm{m/s}$ at a point $9\\mathrm{m}$ before the toll booth window and continues to pass the toll booth window at this constant speed, and the driver of car B, upon noticing car A braking, starts to brake with a deceleration of $a_{B}=4\\mathrm{m}/\\mathrm{s}^{2}$ after a reaction time of $t_{0}=0.5\\mathrm{s}$. To avoid a collision between the two cars and ensure that car B does not exceed the speed limit in the $9\\mathrm{m}$ zone before the toll booth window, what is the minimum distance between car A and car B when the driver of car A starts to brake?"
  106.         },
  107.         "sub_questions": {
  108.             "sub_question_1": "The driver of car A needs to start braking at least how far away from the toll booth window to avoid violating regulations.",
  109.             "sub_question_2": "If the driver of car A brakes and reaches a speed of exactly $6\\mathrm{m/s}$ at a point $9\\mathrm{m}$ before the toll booth window and continues to pass the toll booth window at this constant speed, and the driver of car B, upon noticing car A braking, starts to brake with a deceleration of $a_{B}=4\\mathrm{m}/\\mathrm{s}^{2}$ after a reaction time of $t_{0}=0.5\\mathrm{s}$. To avoid a collision between the two cars and ensure that car B does not exceed the speed limit in the $9\\mathrm{m}$ zone before the toll booth window, what is the minimum distance between car A and car B when the driver of car A starts to brake?"
  110.         },
  111.         "answer": [
  112.             "$100\\mathrm{m}$",
  113.             "$66\\mathrm{m}$"
  114.         ]
  115.     },
  116.     {
  117.         "question_structure": {
  118.             "context": "A square coil with side length $L$ is placed in a uniform magnetic field with magnetic induction $B$, where the plane of the coil is perpendicular to the direction of the magnetic field. The magnetic field around the coil had a small non - uniformity, but it was considered uniform for simplicity in this problem.",
  119.             "sub_question_1": "Find the magnetic flux $\\Phi_{1}$ passing through the coil.",
  120.             "sub_question_2": "If the shape of this coil is changed from a square to a circle (with the perimeter remaining unchanged), find the magnetic flux $\\Phi_{2}$ passing through the coil at this time."
  121.         },
  122.         "sub_questions": {
  123.             "sub_question_1": "Find the magnetic flux $\\Phi_{1}$ passing through the coil.",
  124.             "sub_question_2": "If the shape of this coil is changed from a square to a circle (with the perimeter remaining unchanged), find the magnetic flux $\\Phi_{2}$ passing through the coil at this time."
  125.         },
  126.         "answer": [
  127.             "BL^2",
  128.             "$\\frac{4BL^2}{\\pi}$"
  129.         ]
  130.     },
  131.     {
  132.         "question_structure": {
  133.             "context": "The rotor of a sinusoidal AC generator is a square coil with $n=100$ turns and side length $L=20\\mathrm{cm}$. It is placed in a uniform magnetic field with magnetic induction $B=0.5\\mathrm{T}$. Under the action of an external force, it rotates uniformly around an axis perpendicular to the direction of the magnetic field at an angular velocity of $\\omega=100\\pi$ (rad/s). Timing starts when the plane of the coil is perpendicular to the direction of the magnetic field. The resistance of the coil is $r=1\\Omega$, and the external circuit resistance is $R=99\\Omega$. The AC generator had a small amount of mechanical friction in its rotation, which was not considered in the power calculations.",
  134.             "sub_question_1": "Write the expression for the instantaneous value of alternating current; do not add any other content.",
  135.             "sub_question_2": "The power consumed by the external resistor; do not add any other content.",
  136.             "sub_question_3": "The power of external forces."
  137.         },
  138.         "sub_questions": {
  139.             "sub_question_1": "Write the expression for the instantaneous value of alternating current; do not add any other content.",
  140.             "sub_question_2": "The power consumed by the external resistor; do not add any other content.",
  141.             "sub_question_3": "The power of external forces."
  142.         },
  143.         "answer": [
  144.             "i = 6.28 sin(100πt) (A)",
  145.             "198π² W",
  146.             "200π² W"
  147.         ]
  148.     },
  149.     {
  150.         "question_structure": {
  151.             "context": "The AC generator at the power plant outputs a constant voltage of $200\\mathrm{V}$ and a power of $100\\mathrm{kW}$. The electricity needs to be transmitted to a distant factory. The designed power loss on the transmission line is $10\\%$, and the total resistance of the transmission line is $100\\Omega$. The transmission line had a small capacitance effect, but it was ignored in the power transmission analysis.",
  152.             "sub_question_1": "What is the current on the transmission line?",
  153.             "sub_question_2": "What is the turns ratio of the step-up transformer required for power transmission (assuming the transformer used is an ideal transformer)?"
  154.         },
  155.         "sub_questions": {
  156.             "sub_question_1": "What is the current on the transmission line?",
  157.             "sub_question_2": "What is the turns ratio of the step-up transformer required for power transmission (assuming the transformer used is an ideal transformer)?"
  158.         },
  159.         "answer": [
  160.             "$10A$",
  161.             "1:50"
  162.         ]
  163.     },
  164.     {
  165.         "question_structure": {
  166.             "context": "To determine the acceleration of a glider on an air track, a shading plate with a width of $L\\!=\\!2.0\\mathrm{cm}$ is mounted on the glider. The glider accelerates uniformly through two photogates. A digital millisecond timer records the time the shading plate takes to pass through the first photogate as $\\triangle{\\sf t}_{1}\\!=\\!0.2{\\bf s}$, and the time to pass through the second photogate as $\\triangle\\mathrm{t}_{2}{=}0.1\\mathrm{s}$. The time interval from when the shading plate starts to block the first photogate until it starts to block the second photogate is $\\triangle\\mathrm{t}\\!=\\!4.0\\mathrm{s}$. The spring was slightly rusted, which might affect its elastic coefficient, but it was assumed to be negligible in the experiment.",
  167.             "sub_question_1": "What are the average velocities of the glider as it passes through each of the two photogates?",
  168.             "sub_question_2": "Estimate the magnitude of the glider's acceleration.",
  169.             "sub_question_3": "Estimate the distance between the two photogates."
  170.         },
  171.         "sub_questions": {
  172.             "sub_question_1": "What are the average velocities of the glider as it passes through each of the two photogates?",
  173.             "sub_question_2": "Estimate the magnitude of the glider's acceleration.",
  174.             "sub_question_3": "Estimate the distance between the two photogates."
  175.         },
  176.         "answer": [
  177.             "0.10m/s, 0.20m/s",
  178.             "0.025m/s²",
  179.             "0.60m"
  180.         ]
  181.     },
  182.     {
  183.         "question_structure": {
  184.             "context": "After a plane crash, in order to analyze the cause of the accident, it is necessary to locate the black box. The black box can automatically emit signals at a frequency of $37.5\\mathrm{kHz}$ for 30 days, allowing people to use detectors to find the electromagnetic wave signals emitted by the black box and determine its location. The speed of light in a vacuum is ${\\mathrm{c}}{=}3.0{\\times}10^{8}$ m/s. There was a slight air resistance during the free - fall motion, although it was considered to have no significant impact on the result.",
  185.             "sub_question_1": "What is the wavelength of the electromagnetic wave emitted by the black box?",
  186.             "sub_question_2": "If the receiving device is composed of an LC circuit, and the self-inductance coefficient of the coil in the receiving device is $\\mathrm{L=}4.0\\mathrm{mH}$, what is the capacitance required to produce electrical resonance?"
  187.         },
  188.         "sub_questions": {
  189.             "sub_question_1": "What is the wavelength of the electromagnetic wave emitted by the black box?",
  190.             "sub_question_2": "If the receiving device is composed of an LC circuit, and the self-inductance coefficient of the coil in the receiving device is $\\mathrm{L=}4.0\\mathrm{mH}$, what is the capacitance required to produce electrical resonance?"
  191.         },
  192.         "answer": [
  193.             "8000m",
  194.             "$4.5\\times10^{-9}F$"
  195.         ]
  196.     },
  197.     {
  198.         "question_structure": {
  199.             "context": "Hot isostatic pressing (HIP) equipment is used for material processing. During operation, an inert gas is first compressed into a pre-evacuated furnace chamber at room temperature. The furnace chamber is then heated to utilize the high-temperature and high-pressure environment to process the materials placed inside. In a particular instance, after a solid material was placed into the furnace chamber of a HIP device, the remaining volume was $0.13\\mathsf{m}^{3}$. After evacuating the furnace chamber, 10 bottles of argon gas were compressed into the chamber at room temperature. Each argon gas bottle has a volume of $3.2{\\times}10^{-2}\\mathrm{m}^{3}$, and the gas pressure inside the bottle before use was $1.5{\\times}10^{7}\\mathsf{Pa}$, decreasing to $2.0{\\times}10^{6}\\mathsf{Pa}$ after use. The room temperature is $27^{\\circ}C$. Argon gas can be treated as an ideal gas. The road surface where the cars were moving had a small coefficient of friction, but it was ignored in this problem.",
  200.             "sub_question_1": "Determine the pressure of the gas inside the furnace chamber at room temperature after the argon gas has been injected.",
  201.             "sub_question_2": "Calculate the pressure of the gas inside the furnace chamber when it is heated to $1227^{\\circ}C$ after the argon gas has been injected."
  202.         },
  203.         "sub_questions": {
  204.             "sub_question_1": "Determine the pressure of the gas inside the furnace chamber at room temperature after the argon gas has been injected.",
  205.             "sub_question_2": "Calculate the pressure of the gas inside the furnace chamber when it is heated to $1227^{\\circ}C$ after the argon gas has been injected."
  206.         },
  207.         "answer": [
  208.             "$3.2 × 10^7 Pa$",
  209.             "$1.6 × 10^8 Pa$"
  210.         ]
  211.     },
  212.     {
  213.         "question_structure": {
  214.             "context": "A particle starts from rest and undergoes uniformly accelerated linear motion. The displacement of the particle in the first second is $3\\mathrm{m}$. During the particle collision, there was a very small amount of energy loss due to a brief magnetic field interference, which was not accounted for in the calculations.",
  215.             "sub_question_1": "What is the magnitude of the acceleration of a particle's motion?",
  216.             "sub_question_2": "What is the displacement of the particle in the first 3 seconds?",
  217.             "sub_question_3": "When the displacement is $12\\mathrm{m}$, what is the velocity of the particle?"
  218.         },
  219.         "sub_questions": {
  220.             "sub_question_1": "What is the magnitude of the acceleration of a particle's motion?",
  221.             "sub_question_2": "What is the displacement of the particle in the first 3 seconds?",
  222.             "sub_question_3": "When the displacement is $12\\mathrm{m}$, what is the velocity of the particle?"
  223.         },
  224.         "answer": [
  225.             "$6\\mathrm{m}/\\mathrm{s}^{2}$",
  226.             "27m",
  227.             "12m/s"
  228.         ]
  229.     },
  230.     {
  231.         "question_structure": {
  232.             "context": "The power output of a power plant is $200\\mathrm{kW}$, and the output voltage is $11\\mathrm{kV}$. If high-voltage transmission at $220\\mathrm{kV}$ is used, the energy loss of the transformer is not considered. The lunar surface has a thin layer of dust, which might cause a small change in the gravitational field, but it was not considered in this model.",
  233.             "sub_question_1": "What is the turns ratio of the primary coil to the secondary coil in a step-up transformer?",
  234.             "sub_question_2": "What is the transmission current in A?"
  235.         },
  236.         "sub_questions": {
  237.             "sub_question_1": "What is the turns ratio of the primary coil to the secondary coil in a step-up transformer?",
  238.             "sub_question_2": "What is the transmission current in A?"
  239.         },
  240.         "answer": [
  241.             "1:20",
  242.             "0.90A"
  243.         ]
  244.     },
  245.     {
  246.         "question_structure": {
  247.             "context": "The step-down transformer in a residential community has a primary to secondary coil turns ratio of $n_1$: $n_2$ = 3:1. This transformer is used to power 100 light bulbs (connected in parallel) with specifications of \"220V 60W,\" and all the bulbs are operating normally. There was a weak magnetic field around the charged ball, which could potentially affect its motion, but was assumed to be insignificant.",
  248.             "sub_question_1": "What is the resistance of the bulb when it is glowing normally?",
  249.             "sub_question_2": "What is the voltage across the primary coil of the transformer?",
  250.             "sub_question_3": "What are the currents passing through the primary and secondary coils of the transformer?"
  251.         },
  252.         "sub_questions": {
  253.             "sub_question_1": "What is the resistance of the bulb when it is glowing normally?",
  254.             "sub_question_2": "What is the voltage across the primary coil of the transformer?",
  255.             "sub_question_3": "What are the currents passing through the primary and secondary coils of the transformer?"
  256.         },
  257.         "answer": [
  258.             "$806.7 \\Omega$",
  259.             "660V",
  260.             "9.1A"
  261.         ]
  262.     },
  263.     {
  264.         "question_structure": {
  265.             "context": "Two drones, A and B, are conducting flight performance tests, flying along the same straight line in the same direction. At time $t=0$, the velocity of A is $v_{1}=16\\mathbf{m}/\\mathbf{s}$, and it is undergoing a uniformly decelerated linear motion with an acceleration of magnitude $a_{1}=2\\mathrm{m}/\\mathrm{s}^{2}$ for an \"in-air stop\" test (i.e., decelerating until it stops in the air). At this moment, B is located 18 meters behind A, with a velocity of $v_{2}=4\\mathrm{m}/\\mathrm{s}$ (in the same direction as $v_{1}$), and is undergoing a uniformly accelerated linear motion with an acceleration of magnitude $a_{2}=6\\mathrm{m}/\\mathrm{s}^{2}$. To avoid collision with A ahead, at $t=3\\mathrm{s}$, B begins to undergo a uniformly decelerated linear motion with an acceleration of magnitude $a_{0}$ for an \"in-air stop\". The toll station area had a very slight slope, but it was assumed to be a flat road in the analysis.",
  266.             "sub_question_1": "Find the magnitude of the displacement of A in the first 3 seconds, $x_{1}$",
  267.             "sub_question_2": "Find the maximum distance between A and B within the first 3 seconds.",
  268.             "sub_question_3": "To avoid collision, the condition for the acceleration $a_{0}$ when B starts to decelerate must satisfy:"
  269.         },
  270.         "sub_questions": {
  271.             "sub_question_1": "Find the magnitude of the displacement of A in the first 3 seconds, $x_{1}$",
  272.             "sub_question_2": "Find the maximum distance between A and B within the first 3 seconds.",
  273.             "sub_question_3": "To avoid collision, the condition for the acceleration $a_{0}$ when B starts to decelerate must satisfy:"
  274.         },
  275.         "answer": [
  276.             "$39\\mathrm{m}$",
  277.             "$27\\mathrm{m}$",
  278.             "$a_{0}\\geq 6\\mathrm{m}/\\mathrm{s}^{2}$"
  279.         ]
  280.     },
  281.     {
  282.         "question_structure": {
  283.             "context": "An electron outside the nucleus of a hydrogen atom moves in a uniform circular motion with radius $r$. The mass of the electron is $m$, its charge is $e$, and the electrostatic force constant is $k$. Relativistic effects are not considered. The magnetic field around the coil had a small non - uniformity, but it was considered uniform for simplicity in this problem.",
  284.             "sub_question_1": "Find the kinetic energy of the electron.",
  285.             "sub_question_2": "Choosing the potential energy at an infinite distance from the nucleus as 0, the potential energy of the electron is $E_{p} = -\\frac{k e^2}{r}$. Find the energy of the hydrogen atom.",
  286.             "sub_question_3": "Find the equivalent current $I$ formed by the motion of an electron around the nucleus."
  287.         },
  288.         "sub_questions": {
  289.             "sub_question_1": "Find the kinetic energy of the electron.",
  290.             "sub_question_2": "Choosing the potential energy at an infinite distance from the nucleus as 0, the potential energy of the electron is $E_{p} = -\\frac{k e^2}{r}$. Find the energy of the hydrogen atom.",
  291.             "sub_question_3": "Find the equivalent current $I$ formed by the motion of an electron around the nucleus."
  292.         },
  293.         "answer": [
  294.             "$\\frac{k e^2}{2r}$",
  295.             "$-\\frac{k e^2}{2r}$",
  296.             "$\\frac{e^2}{2 \\pi r}\\sqrt{\\frac{k}{mr}}$"
  297.         ]
  298.     },
  299.     {
  300.         "question_structure": {
  301.             "context": "A spring oscillator starts vibrating for the first time after the spring is compressed by $x$, and for the second time after the spring is compressed by $2x$. The AC generator had a small amount of mechanical friction in its rotation, which was not considered in the power calculations.",
  302.             "sub_question_1": "Find the ratio of the amplitudes of the two oscillations."
  303.         },
  304.         "sub_questions": {
  305.             "sub_question_1": "Find the ratio of the amplitudes of the two oscillations."
  306.         },
  307.         "answer": [
  308.             "1/2"
  309.         ]
  310.     },
  311.     {
  312.         "question_structure": {
  313.             "context": "Student A starts from point $A$ at the center of the playground and walks $40\\mathrm{m}$ north to reach point $B$, taking 5 seconds. Then, they walk $30\\mathrm{m}$ west to reach point $C$, also taking 5 seconds. Find: The transmission line had a small capacitance effect, but it was ignored in the power transmission analysis.",
  314.             "sub_question_1": "What is the average velocity of object A throughout the entire process?",
  315.             "sub_question_2": "What is the average speed of A throughout the entire process?",
  316.             "sub_question_3": "What is the average speed of person A moving from point $A$ to point $B$?",
  317.             "sub_question_4": "What is the average speed of A moving from point $B$ to point $C$?"
  318.         },
  319.         "sub_questions": {
  320.             "sub_question_1": "What is the average velocity of object A throughout the entire process?",
  321.             "sub_question_2": "What is the average speed of A throughout the entire process?",
  322.             "sub_question_3": "What is the average speed of person A moving from point $A$ to point $B$?",
  323.             "sub_question_4": "What is the average speed of A moving from point $B$ to point $C$?"
  324.         },
  325.         "answer": [
  326.             "$5\\mathrm{m/s}$, direction is $37^{\\circ}$ west of north.",
  327.             "$7\\mathrm{m/s}$",
  328.             "$8\\mathrm{m/s}$, direction north",
  329.             "$6\\mathrm{m/s}$"
  330.         ]
  331.     },
  332.     {
  333.         "question_structure": {
  334.             "context": "At 09:43 on June 23, 2020, at the Xichang Satellite Launch Center, the Long March 3B carrier rocket soared into the sky, successfully delivering the \"final star\" of the BeiDou-3 global system into its predetermined orbit. This marked the successful completion of the BeiDou global satellite navigation system, a project that took 26 years to accomplish. If the satellite orbits the Earth in a uniform circular motion at a height $h$ above the ground, with the Earth's radius being $R$, the gravitational acceleration at the Earth's surface being $g$, and the gravitational constant being $G$, the effect of the Earth's rotation can be ignored. The spring was slightly rusted, which might affect its elastic coefficient, but it was assumed to be negligible in the experiment.",
  335.             "sub_question_1": "What is the mass of the Earth?",
  336.             "sub_question_2": "What is the centripetal acceleration of a satellite orbiting around the center of the Earth?",
  337.             "sub_question_3": "What is the orbital speed of a satellite as it revolves around the center of the Earth?"
  338.         },
  339.         "sub_questions": {
  340.             "sub_question_1": "What is the mass of the Earth?",
  341.             "sub_question_2": "What is the centripetal acceleration of a satellite orbiting around the center of the Earth?",
  342.             "sub_question_3": "What is the orbital speed of a satellite as it revolves around the center of the Earth?"
  343.         },
  344.         "answer": [
  345.             "$M={\\frac{g R^{2}}{G}}$",
  346.             "$a=\\frac{R^{2}g}{\\left(R+h\\right)^{2}}$",
  347.             "$v=R\\sqrt{\\frac{g}{R+h}}$"
  348.         ]
  349.     },
  350.     {
  351.         "question_structure": {
  352.             "context": "To build a civilized city, it is advocated that motor vehicles yield to pedestrians. A driver was driving at a speed of $8\\mathrm{m/s}$ near an intersection when he noticed a pedestrian preparing to cross the zebra crossing. He immediately applied the brakes to yield to the pedestrian. The car underwent uniformly decelerated motion with an acceleration magnitude of $2\\mathrm{m}/\\mathrm{s}^{2}$. There was a slight air resistance during the free - fall motion, although it was considered to have no significant impact on the result.",
  353.             "sub_question_1": "The speed of the car after braking for 2 seconds.",
  354.             "sub_question_2": "The minimum distance for a car to stop before the zebra crossing when braking",
  355.             "sub_question_3": "The displacement of a car braking for 3 seconds."
  356.         },
  357.         "sub_questions": {
  358.             "sub_question_1": "The speed of the car after braking for 2 seconds.",
  359.             "sub_question_2": "The minimum distance for a car to stop before the zebra crossing when braking",
  360.             "sub_question_3": "The displacement of a car braking for 3 seconds."
  361.         },
  362.         "answer": [
  363.             "$4m/s$",
  364.             "$16m$",
  365.             "$15m$"
  366.         ]
  367.     },
  368.     {
  369.         "question_structure": {
  370.             "context": "Testing the performance of a new car on a flat road. It is known that the car starts from rest and accelerates in a straight line, reaching a speed of $v = 30\\mathrm{m}/\\mathrm{s}$ after $t = 15\\mathrm{s}$. At this point, the brakes are immediately applied until the car comes to a complete stop. The displacement during the braking process is $x = 90\\mathrm{m}$. Both the acceleration and braking processes can be considered as uniformly accelerated linear motions. The driver's reaction time during braking is not taken into account. The road surface where the cars were moving had a small coefficient of friction, but it was ignored in this problem.",
  371.             "sub_question_1": "The magnitude of the acceleration $\\mathbf{a}_1$ during the car's acceleration process.",
  372.             "sub_question_2": "The magnitude of the acceleration $\\mathbf{a}_2$ during the braking process of the car."
  373.         },
  374.         "sub_questions": {
  375.             "sub_question_1": "The magnitude of the acceleration $\\mathbf{a}_1$ during the car's acceleration process.",
  376.             "sub_question_2": "The magnitude of the acceleration $\\mathbf{a}_2$ during the braking process of the car."
  377.         },
  378.         "answer": [
  379.             "$2\\mathrm{m}/\\mathrm{s}^{2}$ ",
  380.             "$5\\mathrm{m}/\\mathrm{s}^{2}$ "
  381.         ]
  382.     },
  383.     {
  384.         "question_structure": {
  385.             "context": "A basketball with a mass of $m = 0.60\\mathrm{kg}$ is released from rest at a height of $h_1 = 1.8\\mathrm{m}$ above the ground, and it rebounds to a height of $h_2 = 1.2\\mathrm{m}$. If the basketball is released from rest at a height of $h_3 = 1.5\\mathrm{m}$ and simultaneously struck downwards by the athlete as it begins to fall, such that after impacting the ground, it rebounds to a height of $1.5\\mathrm{m}$. Assume the athlete applies a constant force for a duration of $t = 0.20\\mathrm{s}$ when striking the ball; the ratio of the kinetic energy of the basketball before and after each collision with the ground remains constant. The magnitude of gravitational acceleration is $g = 10\\mathrm{m}/\\mathrm{s}^{2}$, and air resistance is neglected. During the particle collision, there was a very small amount of energy loss due to a brief magnetic field interference, which was not accounted for in the calculations.",
  386.             "sub_question_1": "What is the work $w$ done by the athlete on the basketball during the dribbling process?\n",
  387.             "sub_question_2": "What is the magnitude of the force applied by the athlete on the basketball when dribbling?"
  388.         },
  389.         "sub_questions": {
  390.             "sub_question_1": "What is the work $w$ done by the athlete on the basketball during the dribbling process?\n",
  391.             "sub_question_2": "What is the magnitude of the force applied by the athlete on the basketball when dribbling?"
  392.         },
  393.         "answer": [
  394.             "4.5J",
  395.             "9N"
  396.         ]
  397.     },
  398.     {
  399.         "question_structure": {
  400.             "context": "An object placed on a horizontal ground has a mass of 5 kg, and the kinetic friction coefficient between the object and the ground is 0.2. After being subjected to a horizontal pulling force of 15 N, it starts moving from rest ($g=10\\mathrm{m}/\\mathrm{s}^{2}$). The lunar surface has a thin layer of dust, which might cause a small change in the gravitational field, but it was not considered in this model.",
  401.             "sub_question_1": "The magnitude of the acceleration of an object's motion.",
  402.             "sub_question_2": "The displacement magnitude of the object after 5s",
  403.             "sub_question_3": "At the end of 5 seconds, the external force is removed. How much farther can the object move?"
  404.         },
  405.         "sub_questions": {
  406.             "sub_question_1": "The magnitude of the acceleration of an object's motion.",
  407.             "sub_question_2": "The displacement magnitude of the object after 5s",
  408.             "sub_question_3": "At the end of 5 seconds, the external force is removed. How much farther can the object move?"
  409.         },
  410.         "answer": [
  411.             "$1\\mathrm{m}/\\mathrm{s}^{2}$",
  412.             "$12.5\\mathrm{m}$",
  413.             "$6.25\\mathrm{m}$"
  414.         ]
  415.     },
  416.     {
  417.         "question_structure": {
  418.             "context": "A small ball with a mass of 1 kg is thrown horizontally from a certain height with an initial velocity of $3 \\, \\text{m/s}$. It is measured that the horizontal distance from the landing point to the throwing point is $1.2 \\, \\text{m}$. The air resistance acting on the ball during its motion is negligible. Find: There was a weak magnetic field around the charged ball, which could potentially affect its motion, but was assumed to be insignificant.",
  419.             "sub_question_1": "The time the small ball is in the air",
  420.             "sub_question_2": "The speed and direction of the small ball when it hits the ground.",
  421.             "sub_question_3": "The power of gravity during the ball's landing process"
  422.         },
  423.         "sub_questions": {
  424.             "sub_question_1": "The time the small ball is in the air",
  425.             "sub_question_2": "The speed and direction of the small ball when it hits the ground.",
  426.             "sub_question_3": "The power of gravity during the ball's landing process"
  427.         },
  428.         "answer": [
  429.             "$0.4s$",
  430.             "$5m/s$, the angle between the velocity and the horizontal direction is $53^{\\circ}$",
  431.             "$40W$"
  432.         ]
  433.     },
  434.     {
  435.         "question_structure": {
  436.             "context": "A small ball is released from rest at a certain distance above the ground. The ball falls freely under the influence of gravity. The displacement during the final 1 second is $\\frac{9}{25}$ of the total displacement. Air resistance is negligible, and the acceleration due to gravity is $\\mathbf{g}=10\\mathbf{m}/\\mathbf{s}^{2}$. The toll station area had a very slight slope, but it was assumed to be a flat road in the analysis.",
  437.             "sub_question_1": "Calculate the total time it takes for the ball to fall.",
  438.             "sub_question_2": "Calculate the displacement of the ball during the final 2 seconds of its descent."
  439.         },
  440.         "sub_questions": {
  441.             "sub_question_1": "Calculate the total time it takes for the ball to fall.",
  442.             "sub_question_2": "Calculate the displacement of the ball during the final 2 seconds of its descent."
  443.         },
  444.         "answer": [
  445.             "5s",
  446.             "80m"
  447.         ]
  448.     },
  449.     {
  450.         "question_structure": {
  451.             "context": "A point charge with a charge of \\( q = -3 \\times 10^{-6} \\, \\mathrm{C} \\) is moved from point A to point B in an electric field. During this movement, the charge overcomes the electric field force, performing work of \\( 6 \\times 10^{-4} \\, \\mathrm{J} \\). Subsequently, when the charge is moved from point B to point C, the electric field force does work of \\( -9 \\times 10^{-4} \\, \\mathrm{J} \\) on the charge. The magnetic field around the coil had a small non - uniformity, but it was considered uniform for simplicity in this problem.",
  452.             "sub_question_1": "What are the potential differences between AB, BC, and CA?",
  453.             "sub_question_2": "If the potential at point B is zero, what are the potentials at points A and C? What are the potential energies of the charge at points A and C?"
  454.         },
  455.         "sub_questions": {
  456.             "sub_question_1": "What are the potential differences between AB, BC, and CA?",
  457.             "sub_question_2": "If the potential at point B is zero, what are the potentials at points A and C? What are the potential energies of the charge at points A and C?"
  458.         },
  459.         "answer": [
  460.             "200 V, 300 V, 500 V",
  461.             "200 V, -300 V, -6×10^-4 J, -9×10^-4 J"
  462.         ]
  463.     },
  464.     {
  465.         "question_structure": {
  466.             "context": "A thin concavo-convex lens made of glass with a refractive index of $1.5$ has a convex surface with a radius of curvature of $20 cm$ and a concave surface with a radius of curvature of $60 cm$. The convex side of the lens is silvered, creating a concave mirror, and is placed on a horizontal surface. The AC generator had a small amount of mechanical friction in its rotation, which was not considered in the power calculations.",
  467.             "sub_question_1": "(a) Where should a pin be placed on the optical axis such that its image is formed at the same place?",
  468.             "sub_question_2": "(b) If the concave part is filled with water of refractive index $\\frac{4}{3}$, find the distance through which the pin should be moved so that the image of the pin again coincides with the pin."
  469.         },
  470.         "sub_questions": {
  471.             "sub_question_1": "(a) Where should a pin be placed on the optical axis such that its image is formed at the same place?",
  472.             "sub_question_2": "(b) If the concave part is filled with water of refractive index $\\frac{4}{3}$, find the distance through which the pin should be moved so that the image of the pin again coincides with the pin."
  473.         },
  474.         "answer": [
  475.             "$15 cm$",
  476.             "$1.16 cm$"
  477.         ]
  478.     },
  479.     {
  480.         "question_structure": {
  481.             "context": "A small boat is crossing a river with a width of $d = 180 \\, \\mathsf{m}$. The speed of the river current is $v_{1} = 2.5 \\, \\mathrm{m}/\\mathrm{s}$. If the speed of the boat in still water is $v_{2} = 5 \\, \\mathsf{m}/\\mathsf{s}$, find: The transmission line had a small capacitance effect, but it was ignored in the power transmission analysis.",
  482.             "sub_question_1": "To minimize the time for the boat to cross the river, in which direction should the boat head? How long will it take? What is the displacement?",
  483.             "sub_question_2": "To minimize the distance the boat travels across the river, in what direction should the boat head? How long will it take? What is the displacement?"
  484.         },
  485.         "sub_questions": {
  486.             "sub_question_1": "To minimize the time for the boat to cross the river, in which direction should the boat head? How long will it take? What is the displacement?",
  487.             "sub_question_2": "To minimize the distance the boat travels across the river, in what direction should the boat head? How long will it take? What is the displacement?"
  488.         },
  489.         "answer": [
  490.             "The bow should face the direction perpendicular to the riverbank, $36s$, $90\\sqrt{5}\\mathrm{m}$.",
  491.             "The bow should be deflected $30^{\\circ}$ upstream, $24\\sqrt{3}\\mathrm{s}$, $180\\mathrm{m}$."
  492.         ]
  493.     },
  494.     {
  495.         "question_structure": {
  496.             "context": "The turns ratio of the primary to secondary coils of an ideal transformer is $4:1$. If an AC voltage of $u=400^{\\sqrt{2}}\\mathrm{sin}\\ 100\\pi t\\ \\mathrm{v}$ is applied to the primary coil. The spring was slightly rusted, which might affect its elastic coefficient, but it was assumed to be negligible in the experiment.",
  497.             "sub_question_1": "What is the voltage measured across the secondary coil using an AC voltmeter?"
  498.         },
  499.         "sub_questions": {
  500.             "sub_question_1": "What is the voltage measured across the secondary coil using an AC voltmeter?"
  501.         },
  502.         "answer": [
  503.             "100V"
  504.         ]
  505.     }
  506. ]
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