[{"problem_text": " An automobile with a mass of $1000 \\mathrm{~kg}$, including passengers, settles $1.0 \\mathrm{~cm}$ closer to the road for every additional $100 \\mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \\mathrm{~km} / \\mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \\mathrm{~cm}$ and $20 \\mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \\mathrm{~m}$. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.\r\n", "answer_latex": " -0.16", "answer_number": "-0.16", "unit": " $ \\mathrm{~mm}$", "source": "class", "problemid": " Problem 3.40", "comment": " ", "solution": ""}, {"problem_text": "Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.", "answer_latex": " $2 \\sqrt{2} \\sin \\frac{\\pi}{2 \\sqrt{2}}$", "answer_number": "2.534324263", "unit": "", "source": "class", "problemid": " Problem 6.14", "comment": " ", "solution": ""}, {"problem_text": "In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \\mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?", "answer_latex": " 210", "answer_number": "210", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": "Problem 2.6 ", "comment": " ", "solution": ""}, {"problem_text": "Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to $1 / e$ of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.\r\n", "answer_latex": " $\\frac{8 \\pi}{\\sqrt{64 \\pi^2+1}}$", "answer_number": "0.9992093669", "unit": "", "source": "class", "problemid": " Problem 3.44", "comment": " ", "solution": ""}, {"problem_text": "What is the minimum escape velocity of a spacecraft from the moon?", "answer_latex": " 2380", "answer_number": "2380", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.28", "comment": " ", "solution": ""}, {"problem_text": "A rocket has an initial mass of $7 \\times 10^4 \\mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\\mathrm{kg} / \\mathrm{s}$. The exhaust velocity is $2500 \\mathrm{~m} / \\mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off?", "answer_latex": "25", "answer_number": "25", "unit": "$\\mathrm{~s}$ ", "source": "class", "problemid": " Problem 9.60", "comment": " ", "solution": ""}, {"problem_text": "A spacecraft of mass $10,000 \\mathrm{~kg}$ is parked in a circular orbit $200 \\mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\\tau=24 \\mathrm{hr}$ )?", "answer_latex": " 2.57", "answer_number": "2.57", "unit": "$10^{11} \\mathrm{~J}$ ", "source": "class", "problemid": " Problem 8.42", "comment": " ", "solution": ""}, {"problem_text": "A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \\mathrm{~s}$ to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?\r\n", "answer_latex": "13.2", "answer_number": "13.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.4", "comment": " ", "solution": ""}, {"problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \\mathrm{~km} / \\mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\\mathrm{LAB}$ angle $\\psi=10^{\\circ}$, what is the final speed of the deuteron?", "answer_latex": "14.44", "answer_number": "14.44", "unit": "$\\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 9.22", "comment": " ", "solution": ""}, {"problem_text": "A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \\mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \\mathrm{~m} / \\mathrm{s}$, find the height of the building, neglecting air resistance.", "answer_latex": " 71", "answer_number": "71", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.30", "comment": " ", "solution": ""}, {"problem_text": "A steel ball of velocity $5 \\mathrm{~m} / \\mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?", "answer_latex": " $4.3$", "answer_number": "4.3", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.42", "comment": " ", "solution": ""}, {"problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \\mathrm{~cm}$ and the mass be $200 \\mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ", "answer_latex": " 35.2", "answer_number": "35.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.18", "comment": " ", "solution": ""}, {"problem_text": "A child slides a block of mass $2 \\mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\\mathrm{m} / \\mathrm{s}$ and the block hits a spring with spring constant $6 \\mathrm{~N} / \\mathrm{m}$, what is the maximum compression of the spring? ", "answer_latex": " 2.3", "answer_number": "2.3", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.26", "comment": " ", "solution": ""}, {"problem_text": "An Earth satellite has a perigee of $300 \\mathrm{~km}$ and an apogee of $3,500 \\mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\\circ}$ around Earth from perigee?", "answer_latex": "1590", "answer_number": "1590", "unit": "$\\mathrm{~km}$ ", "source": "class", "problemid": " Problem 8.24", "comment": " ", "solution": ""}, {"problem_text": "Two masses $m_1=100 \\mathrm{~g}$ and $m_2=200 \\mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \\mathrm{~N} / \\mathrm{m}$. Find the frequency of oscillatory motion for this system.", "answer_latex": " 2.74", "answer_number": "2.74", "unit": "$\\mathrm{rad} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 3.6", "comment": " ", "solution": ""}, {"problem_text": "Calculate the minimum $\\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun. ", "answer_latex": " 5275", "answer_number": "5275", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.38", "comment": " ", "solution": ""}, {"problem_text": "A potato of mass $0.5 \\mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \\mathrm{~s}^{-1}$. ", "answer_latex": " 1000", "answer_number": "1000", "unit": "$\\mathrm{~m} / \\mathrm{s}$  ", "source": "class", "problemid": " Problem 2.54", "comment": " ", "solution": ""}, {"problem_text": "The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?", "answer_latex": " -2", "answer_number": "-2", "unit": "m ", "source": "class", "problemid": "Problem 1.40 ", "comment": " ", "solution": ""}, {"problem_text": "Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\\circ} \\mathrm{N}$, and the lead fell a distance of $27 \\mathrm{~m}$. How far did the lead bullets land from the direct vertical?", "answer_latex": "2.26", "answer_number": "2.26", "unit": " $\\mathrm{~mm}$", "source": "class", "problemid": "Problem 10.22", "comment": " ", "solution": ""}, {"problem_text": "A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \\mathrm{dyne} / \\mathrm{cm}$. The mass is displaced $3 \\mathrm{~cm}$ and released from rest. Calculate the natural frequency $\\nu_0$.", "answer_latex": " 6.9", "answer_number": "6.9", "unit": " $10^{-2} \\mathrm{~s}^{-1}$", "source": "class", "problemid": " Problem 3.2", "comment": " ", "solution": ""}, {"problem_text": "Use the function described in Example 4.3, $x_{n+1}=\\alpha x_n\\left(1-x_n^2\\right)$ where $\\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Determine the lowest value of $n$ for which the two values diverge by more than $30 \\%$.", "answer_latex": " 30", "answer_number": "30", "unit": " ", "source": "class", "problemid": " Problem 4.14", "comment": " ", "solution": ""}, {"problem_text": "A gun fires a projectile of mass $10 \\mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \\mathrm{~m} / \\mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \\mathrm{~m}$ away? Compare the results with those for the case of no retardation.", "answer_latex": " 17.4", "answer_number": "17.4", "unit": "$^{\\circ}$ ", "source": "class", "problemid": "Problem 2.20 ", "comment": " ", "solution": ""}, {"problem_text": "A spacecraft is placed in orbit $200 \\mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ", "answer_latex": " 3.23", "answer_number": "3.23", "unit": " $ \\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 8.30", "comment": " ", "solution": ""}, {"problem_text": "Find the value of the integral $\\int_S(\\nabla \\times \\mathbf{A}) \\cdot d \\mathbf{a}$ if the vector $\\mathbf{A}=y \\mathbf{i}+z \\mathbf{j}+x \\mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \\geq 0$.", "answer_latex": "$-\\pi$", "answer_number": "-3.141592", "unit": "", "source": "class", "problemid": "Problem 1.38", "comment": " ", "solution": ""}, {"problem_text": "A skier weighing $90 \\mathrm{~kg}$ starts from rest down a hill inclined at $17^{\\circ}$. He skis $100 \\mathrm{~m}$ down the hill and then coasts for $70 \\mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ", "answer_latex": "0.18", "answer_number": "0.18", "unit": " ", "source": "class", "problemid": " Problem 2.24", "comment": " ", "solution": ""}, {"problem_text": "Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\\operatorname{Sun}$ is $\\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\\beta<1$, show that the time the comet spends within the orbit of Earth is given by\r\n$$\r\n\\sqrt{2(1-\\beta)} \\cdot(1+2 \\beta) / 3 \\pi \\times 1 \\text { year }\r\n$$\r\nIf the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?", "answer_latex": " 76", "answer_number": "76", "unit": "$ \\text { days }$ ", "source": "class", "problemid": " Problem 8.12", "comment": " ", "solution": ""}, {"problem_text": "A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\\circ} \\mathrm{S}$. If the shells are fired at $37^{\\circ}$ elevation with a speed of $800 \\mathrm{~m} / \\mathrm{s}$, by how much do the shells miss their target?", "answer_latex": " 260", "answer_number": "260", "unit": " $\\mathrm{~m}$", "source": "class", "problemid": " Problem 10.18", "comment": " ", "solution": ""}, {"problem_text": "Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?\r\n", "answer_latex": " 9", "answer_number": "9", "unit": "$10^7 \\mathrm{yr}$ ", "source": "class", "problemid": " Problem 8.46", "comment": " ", "solution": ""}, {"problem_text": "To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \\mathrm{~m} / \\mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?", "answer_latex": "273", "answer_number": "273", "unit": " $\\mathrm{~s}$", "source": "class", "problemid": " Problem 9.62", "comment": " ", "solution": ""}, {"problem_text": "In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\\mathbf{u}_1$ and $\\mathbf{u}_2=\\alpha \\mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\\alpha$ is positive. ", "answer_latex": " $3 \\pm 2 \\sqrt{2}$", "answer_number": "5.828427125", "unit": " ", "source": "class", "problemid": " Problem 9.36", "comment": " ", "solution": ""}, {"problem_text": "Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \\mathrm{~m} / \\mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \\mathrm{~kg}$, including a pressurized tank of mass $10 \\mathrm{~kg}$. The tank includes only $2 \\mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \\mathrm{~m} / \\mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?", "answer_latex": "11", "answer_number": "11", "unit": "$ \\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.12", "comment": " ", "solution": ""}, {"problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9$ km / s collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a LAB angle $\\psi = 10^\\circ$, the final speed of the deuteron is $v_d = 14.44$ km / s and the final speed of the neutron is $v_n = 5.18$ km / s. Another set of solutions for the final speed is $v_d = 5.12$ km / s for the deuteron and $v_n = 19.79$ km / s for the neutron. What is the maximum possible scattering angle of the deuteron?", "answer_latex": "$74.8^\\circ$, $5.2^\\circ$", "answer_number": "30", "unit": "$^\\circ$", "source": "class", "problemid": "9.22 B. ", "comment": ""}, {"problem_text": "A steel ball of velocity $5$ m/s strikes a smooth, heavy steel plate at an angle of $30^\\circ$ from the normal. If the coefficient of restitution is 0.8, at what angle from the normal does the steel ball bounce off the plate?", "answer_latex": "$36^\\circ$", "answer_number": "36", "unit": "$^\\circ$", "source": "class", "problemid": "9.42 B. ", "comment": " "}, {"problem_text": "A string is set into motion by being struck at a point $L/4$ from one end by a triangular hammer. The initial velocity is greatest at $x = L/4$ and decreases linearly to zero at $x = 0$ and $x = L/2$. The region $L/2 \\leq x \\leq L$ is initially undisturbed. Determine the subsequent motion of the string. How many decibels down from the fundamental are the second harmonics?'", "answer_latex": "4.4, 13.3", "answer_number": "4.4", "unit": " dB", "source": "class", "problemid": "13.6 ", "comment": " "}, {"problem_text": "In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. To not hit the cattle, what is the largest time error she could make while pushing the bales out of the airplane? Ignore air resistance.", "answer_latex": "0.68 ", "answer_number": "0.68", "unit": "seconds ", "source": "class", "problemid": " 2.6 B.", "comment": " "}, {"problem_text": "A free neutron is unstable and decays into a proton and an electron. How much energy other than the rest energies of the proton and electron is available if a neutron at rest decays? (This is an example of nuclear beta decay. Another particle, called a neutrino-- actually an antineutrino $\\bar v$ is also produced.)", "answer_latex": " 0.8", "answer_number": "0.8", "unit": "$MeV$ ", "source": "class", "problemid": " 14.30", "comment": " "}, {"problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, assuming the density of air is constant, the maximum height reached is 890 km. Now also include the fact that the acceleration of gravity decreases as the object soars above Earth. Find the height reached.", "answer_latex": "950 ", "answer_number": "950", "unit": "km ", "source": "class", "problemid": "9.64 C. ", "comment": " "}, {"problem_text": "Calculate the effective gravitational field vector $g$ at Earth's surface at the poles. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. How well does the result agree with the difference calculated with the result $g = 9.780356[1 + 0.0052885sin^2\\lambda - 0.0000059 sin^2 (2\\lambda )]$ $m/s^2$ where $\\lambda$ is the latitude?", "answer_latex": " 9.832", "answer_number": "9.832", "unit": "$m/s^2$ ", "source": "class", "problemid": " 10.20", "comment": " "}, {"problem_text": "In nuclear and particle physics, momentum is usually quoted in $MeV / c$ to facilitate calculations. Calculate the kinetic energy of an electron if it has a momentum of $1000$ $MeV/c$", "answer_latex": "$T_{electron} = 999.5$, $T_{proton} = 433$", "answer_number": "999.5", "unit": " $MeV$", "source": "class", "problemid": " 14.32", "comment": " "}, {"problem_text": "A skier weighing $90$ kg starts from rest down a hill inclined at $17^\\circ$. He skis 100 m down the hill and then coasts for 70 m along level snow until he stops. Given a coefficient of kinetic friction of $\\mu_k = 0.18$, what velocity does the skier have at the bottom of the hill?", "answer_latex": "15.6 ", "answer_number": "15.6", "unit": "$m/s$ ", "source": "class", "problemid": " 2.24 B.", "comment": "Uses answer from part A. (coefficient of kinetic friction) "}, {"problem_text": "A rocket starts from rest in free space by emitting mass. At what fraction of the initial mass is the momentum a maximum?", "answer_latex": "$e^{-1}$ ", "answer_number": "0.367879", "unit": " ", "source": "class", "problemid": " 9.54", "comment": " "}, {"problem_text": "A particle moves in a plane elliptical orbit described by the position vector $r = 2b \\sin \\omega ti + b \\cos \\omega tj$ \r\n\r\nWhat is the angle between $v$ and $a$ at time $t = \\frac{\\pi}{2\\omega}$ ?", "answer_latex": "$90^\\circ$", "answer_number": "90", "unit": "$^\\circ$", "source": "class", "problemid": " 1.10 B.", "comment": " "}, {"problem_text": "An Earth satellite has a perigee of $300$ km and an apogee of $3,500$ km above Earth's surface. How far is the satellite above Earth when it has moved halfway from perigee to apogee?", "answer_latex": "1900 ", "answer_number": "1900", "unit": "$km$ ", "source": "class", "problemid": "8.24 (b) ", "comment": " "}, {"problem_text": "In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of $8.5$ N-s. Assume the total rocket mass at launch is $54$ g and that it has a rocket engine of mass $20$ g that burns evenly for $1.5$ s. The rocket diameter is $24$ mm. Assume a constant burn rate of the propellent mass ($11$ g), a rocket exhaust speed $800$ m/s, vertical ascent, and drag coefficient $c_w = 0.75$. Take into account the change of rocket mass with time and omit the effect of gravity. The rocket's speed at burn out is 131 m/s. How far has the rocket traveled at that moment?", "answer_latex": " 108", "answer_number": "108", "unit": "m ", "source": "class", "problemid": "9.66 B. ", "comment": " "}, {"problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9$ km / s collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a LAB angle $\\psi = 10^\\circ$, what is the final speed of the neutron?", "answer_latex": " 5.18", "answer_number": "5.18", "unit": "km / s ", "source": "class", "problemid": "9.22 A. ", "comment": "9.22 A, but only the neutron (not deuteron, which has already been annotated) "}, {"problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, determine the maximum height reached. Assume the density of air is constant.", "answer_latex": " 890", "answer_number": "890", "unit": "km ", "source": "class", "problemid": "9.64 B. ", "comment": " "}, {"problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, assuming the density of air is constant, the maximum height reached is 890 km. Including the fact that the acceleration of gravity decreases as the object soars above Earth, the height reached is 950 km. Now add the effects of the decrease in air density with altitude to the calculation. We can very roughly represent the air density by $log_{10}(\\rho) = -0.05h + 0.11$ where $\\rho$ is the air density in $kg/m^3$ and $h$ is the altitude above Earth in km. Determine how high the object now goes.", "answer_latex": "8900 ", "answer_number": "8900", "unit": "km ", "source": "class", "problemid": "9.64 D. ", "comment": " "}, {"problem_text": "A racer attempting to break the land speed record rockets by two markers spaced $100$ m apart on the ground in a time of $0.4$ $\\mu s$ as measured by an observer on the ground. How far apart do the two markers appear to the racer? ", "answer_latex": "55.3 ", "answer_number": "55.3", "unit": "$m$ ", "source": "class", "problemid": " 14.12", "comment": "Omitted the last two questions / answers "}, {"problem_text": "A billiard ball of intial velocity $u_1$ collides with another billard ball (same mass) initially at rest. The first ball moves off at $\\psi = 45^\\circ$. For an elastic collision, say the velocities of both balls after the collision is $v_1 = v_2 = \\frac{u_1}{\\sqrt(2)}$. At what LAB angle does the second ball emerge?", "answer_latex": "45 ", "answer_number": "45", "unit": "$^\\circ$", "source": "class", "problemid": "9.34 B. ", "comment": " "}, {"problem_text": "Calculate the effective gravitational field vector $\\textbf{g}$ at Earth's surface at the equator. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. ", "answer_latex": " 9.780", "answer_number": "9.780", "unit": "$m/s^2$ ", "source": "class", "problemid": " 10.20 B.", "comment": "Only calculating the equator, not the polar."}, {"problem_text": "An astronaut travels to the nearest star system, 4 light years away, and returns at a speed $0.3c$. How much has the astronaut aged relative to those people remaining on Earth?", "answer_latex": "25.4", "answer_number": "25.4", "unit": "years ", "source": "class", "problemid": " 14.20", "comment": "Astronaut ages 25.4, people on Earth age 26.7 "}, {"problem_text": "In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of $8.5$ N-s. Assume the total rocket mass at launch is $54$ g and that it has a rocket engine of mass $20$ g that burns evenly for $1.5$ s. The rocket diameter is $24$ mm. Assume a constant burn rate of the propellent mass ($11$ g), a rocket exhaust speed $800$ m/s, vertical ascent, and drag coefficient $c_w = 0.75$. Take into account the change of rocket mass with time and omit the effect of gravity. Find the rocket's speed at burn out.", "answer_latex": "131 ", "answer_number": "131", "unit": "m/s ", "source": "class", "problemid": "9.66 A. ", "comment": " "}, {"problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglect air resistance and assume that the acceleration of gravity is constant. Determine how high the launched object can reach above the surface of Earth.", "answer_latex": "3700 ", "answer_number": "3700", "unit": "km ", "source": "class", "problemid": "9.64 A. ", "comment": " "}, {"problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_w = 0.5$, the softball radius be $5$ cm and the mass be $200$ g. Given a speed of 35.2 m/s, find the initial elevation angle that allows the ball to most easily clear the fence.", "answer_latex": " $40.7^\\circ$", "answer_number": "40.7", "unit": "$^\\circ$", "source": "class", "problemid": " 2.18 B.", "comment": "Used answer given in part A. for speed"}, {"problem_text": "Show that the small angular deviation of $\\epsilon$ of a plumb line from the true vertical (i.e., toward the center of Earth) at a point on Earth's surface at a latitude $\\lambda$ is $\\epsilon = \\frac{R\\omega^2sin\\lambda cos\\lambda}{g_0 - R\\omega^2 cos^2\\lambda}$ where R is the radius of Earth. What is the value (in seconds of arc) of the maximum deviation? Note that the entire denominator in the answer is actually the effective $g$, and $g_0$ denotes the pure gravitational component.", "answer_latex": "6", "answer_number": "6", "unit": "min", "source": "class", "problemid": " 10.12", "comment": " "}, {"problem_text": "A potato of mass 0.5 kg moves under Earth's gravity with an air resistive force of -$kmv$. The terminal velocity of the potato when released from rest is $v = 1000$ m/s, with $k=0.01s^{-1}$. Find the maximum height of the potato if it has the same value of k, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120$ m/s.", "answer_latex": "680 ", "answer_number": "680", "unit": "$m$ ", "source": "class", "problemid": " 2.54 B.", "comment": "Used terminal velocity answer of 1000 m/s from part A. "}]