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MATINS BETERE 1) If an object is in Earth's orbit (a=1.000 A.U., e=0.0167) and suddenly increases its energy by 10 per cent without increasing its angular momentum [perhaps it fires a

rocket engine towards the Sun, accelerating it directly outwards]: what is the final semi-major axis and eccentricity of the object's orbit? Note: E<0. Therefore increasing E by 10% means less negative or 90% of the original value. (Marks: 4) 2) The Earth was last at perihelion on January 3, 2022. The period of the Earth's orbit is 365.25 days. (a) If the Earth's orbit was a perfect circle when (give the actual calendar date in your answer!) would it reach a True Anomaly (angle with respect to perihelion) of 160 degrees? (b) But given that the Earth's orbit is slightly elliptical (e-0.0167), when (calendar date) will the Earth actually be at True Anomaly of 160 degrees? (c) How far will the Earth be from the Sun at that time? (d) How fast will the Earth be moving in its orbit on that day? (e) (for comparison to (c) and (d)) what is the distance (from the Sun) and speed of the Earth when it is at perihelion? (Marks 10) 3) How bright will Pluto appear in reflected or scattered sun-light (total Flux received in W/m²) at the Earth when Pluto is at Opposition and a distance from the Sun of 40.0 A.U.? Assume that Pluto has an albedo of 0.60 and a radius of 1.2 x 10³ km. (Marks: 6) DELL

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The University of Queensland, School of Mechanical and Mining Engineering MECH2210 Dynamics and Orbital Mechanics Section B1- Tutorial Problems 2022 Assignment B1, Submit: 0.3 & 17 due Mon wk 11 Online Dynamics Revision Quiz The following problems cover orbital mechanics material - please attempt all: Module/Question - 1/7, 2/12, 2/21, 3/5-12, 4/5, 6/3. (Answers online) F ᏛᎾ . B m Problem 1 Explorer No.1 launched in January 1958 had perigee and apogee heights above the Earth's surface of 360 km and 2550 km and an inclination of 33.2°. Calculate the: a) orbit period, b) its eccentricity, c) the total energy (per unit mass of the orbit) d) the angular momentum (per unit mass of the orbit), e) the maximum and minimum speeds and orbit positions f) the minimum impulse to change the inclination to 0° assuming the satellite passes the equator at its perigee, g) the minimum impulse Av required to escape Earth. (Answers: 1.92hr,0.14, -25.4 MJ/kg, 5.53x1010 m²/s, 8.2p & 6.2a km/s, 3.54 km/s, 2.67 km/s) Problem 2 An earth satellite is tracked from ground stations and is observed to have an altitude of 2200 km, a velocity magnitude 7 km/s and a radial velocity of 2.7 km/s. Determine: a) the total orbital energy per unit mass b) the orbital eccentricity e c) the minimum altitude and the maximum speed and d) the true anomalies at the observed position and at the point of maximum speed. e) Identify any potential difficulties with the orbit. (Answers: -22 MJ/kg, 0.389, -835km & 10 km/s, 105.3deg & 0 deg, crash into Earth) Problem 3 An early warning defence system detects an UFO travelling at 800km above earth's surface with a horizontal (perpendicular) speed of 8 km/s and a vertical (radial) speed of 1.3 km/s. Determine: a) the total energy (per unit mass of the orbit) b) the angular momentum (per unit mass of the orbit), c) the eccentricity of the orbit e d) the minimum and maximum radii of the orbit e) the true anomaly 9 at the point of observation. f) whether the UFO is possibly a missile, Earth satellite or comet and explain your answer. g) the minimum measured speed v of the UFO for it to be a comet and explain why. (Tot. 21 mks = 3 x 7) Problem 4


Problem 2 An earth satellite is tracked from ground stations and is observed to have an altitude of 2200 km, a velocity magnitude 7 km/s and a radial velocity of 2.7 km/s. Determine: a) the total orbital energy per unit mass b) the orbital eccentricity e c) the minimum altitude and the maximum speed and d) the true anomalies at the observed position and at the point of maximum speed. e) Identify any potential difficulties with the orbit. (Answers: -22 MJ/kg, 0.389, -835km & 10 km/s, 105.3deg & 0 deg, crash into Earth)


1. (10 pts) Manually draw the layout and patterns of motion of the solar system. Please include the Sun and all planets. Draw the orbits of all planets around the Sun (2 pts) Label the orbit direction and spin direction of all planets. Label the spin direction of the Sun. (2 pts) Label the planets' average orbital distances from the Sun in AU and their orbital periods in Earth years (2 pts) Label the size of the Sun and each planet as compared to Earth's radius (e.g. 0.5 Earth radius, 2 Earth radii,...) (2 pts) Point out the exceptions to the patterns of motion (2 pts). (The sizes and distances do not need to be drawn to scale, which is impossible to do on letter sized paper anyway.) • •


MATINS BETERE 1) If an object is in Earth's orbit (a=1.000 A.U., e=0.0167) and suddenly increases its energy by 10 per cent without increasing its angular momentum [perhaps it fires a rocket engine towards the Sun, accelerating it directly outwards]: what is the final semi-major axis and eccentricity of the object's orbit? Note: E<0. Therefore increasing E by 10% means less negative or 90% of the original value. (Marks: 4) 2) The Earth was last at perihelion on January 3, 2022. The period of the Earth's orbit is 365.25 days. (a) If the Earth's orbit was a perfect circle when (give the actual calendar date in your answer!) would it reach a True Anomaly (angle with respect to perihelion) of 160 degrees? (b) But given that the Earth's orbit is slightly elliptical (e-0.0167), when (calendar date) will the Earth actually be at True Anomaly of 160 degrees? (c) How far will the Earth be from the Sun at that time? (d) How fast will the Earth be moving in its orbit on that day? (e) (for comparison to (c) and (d)) what is the distance (from the Sun) and speed of the Earth when it is at perihelion? (Marks 10) 3) How bright will Pluto appear in reflected or scattered sun-light (total Flux received in W/m²) at the Earth when Pluto is at Opposition and a distance from the Sun of 40.0 A.U.? Assume that Pluto has an albedo of 0.60 and a radius of 1.2 x 10³ km. (Marks: 6) DELL


1. You have a glass with mass m bouncing off the floor. Its duration of contact with the floor the time difference between first touching the floor and last touching the floor- is T. Its velocity when it first touches the floor is Viy, and its velocity as it launches off the floor is ufy. If, at any time during its contact with the floor, the force on the glass ever exceeds Fmax, even for the tiniest fraction of a second, the glass will break. Note that the force on the glass during its collision with the floor will not be constant, and you have no idea what the exact force graph looks like. (a) Write down an inequality describing the condition under which it is certain that the glass will break. Hint: Sketching an F vs. t graph might help you think about this. (b) Let's say the inequality you wrote down does not hold. Does this mean that the glass is certain to remain unbroken? Explain.


Suppose you are sitting in a closed room that is magically transported off Earth so that, as shown in the diagram, you are accelerating through space at 9.8 m/s2. According to the equivalence principle, how will you know that you've left Earth?


3. The energy density of photons in the frequency range (v, v+ dv), is given by the blackbody or Planck function: \varepsilon(\nu) d \nu=\frac{8 \pi h}{c^{3}} \frac{v^{3} d \nu}{\exp (h v / k T)-1} a.Derive that the peak of this function occurs at an energy hv=2.82 kT. What isthis relation or law usually called in the literature? b. Derive that, integrated over all frequencies, the energy density is equal to: \begin{array}{l} \varepsilon_{\gamma}=\alpha T^{4} \quad \text { where } \\ \alpha=\frac{\pi^{2}}{15} \frac{k^{4}}{\hbar^{3} c^{3}}=7.56 \times 10^{-16} \mathrm{~J} \mathrm{~m}^{-3} \mathrm{~K}^{-4} \end{array} What is this relation generally known as? c. Hence, calculate the total energy density of the Cosmic MicrowaveBackground (assume T = 2.725 K) and its photon density.


3.Consider the line element of the sphere of radius a:ds2 =a2(d02+sin20do2).The only non-vanishing Christoffel symbols arer=-sin0 cos0,=tana)Write down the metric and the inverse metric,and use the definitionTP (ngva+9uo-09m)=TPvuto reproduce the results written above for and[You can also check that the otherChristoffel symbols vanish,for practice,but this will not be marked.b)Write down the two components of the geodesic equation.b)The geodesics of the sphere are great circles.Thinking of 0 =0 as the North pole and 0=as the South pole,find a set a solutions to the geodesic equation corresponding to meridians,andalso the solution corresponding to the equator.


33 A satellite is orbiting at a distance of 4.2 x 106 m from the surface of the Earth. The radius of the Earth is 6.4 x 106 m. What is the ratio of gravitational force on the satellite in orbit/gravitational force on the satellite on the surface of the Earth? A 0.36 B 0.42 C 0.51 D 0.64 Answer give D


Problem 1. (15 points) Ptolemy vs Copernicus (1) In this problem and the next we will compare how Ptolemy and Copernicus handled the inner (or inferior) and outer (or superior) planets in their respective models of the Solar System. To keep things simple, we will neglect the planets' eccentricities for the purposes of these two problems. (a) As we discussed in class, in Ptolemy's geocentric model of the Solar System the centers of the epicycles for the inferior planets Mercury and Venus are tied to the motion of the Sun, in order to keep these two planets from 'wandering' too far away from the Sun in the sky. The maximum elongations (i.e., angular distance from the Sun) observed for Mercury and Venus are max 22.8° and 46.3°, respectively. Demonstrate that, in the Ptolemaic model, 0 max can be used to estimate the ratio of the radius of the epicycle, E for each planet to that of its deferent, D, but not their absolute values nor their ratios to the Earth-Sun distance. = (b) In the Copernican model, on the other hand, show that the orbital radius of an inferior planet is given by r = R sin max, where R is the Earth's orbital radius, or 1 Astronomical Unit (AU). Use the values of max given above to evaluate r for Mercury and Venus. Give your answers to 3 decimal places. (c) With the help of a diagram, calculate the minimum and maximum distances from the Earth for an inferior planet in the Ptolemaic model in terms of D and the parameter (max. (d) Do the same for the Copernican model, giving your results in terms of R and max. Show that the ratio of minimum to maximum distance is the same for both models. (Do this analytically, rather than numerically.) In this sense, the two models are equivalent, but Copernicus considered the inability to relate the orbital radii of Mercury & Venus to that of the Earth to be a major failing of the old model. Do you agree?