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--- repository:satellite_orbit [2020/02/25 18:47]
porcaro1 [Answer Key]
+++ repository:satellite_orbit [2020/05/04 20:58] (current)
tallpaul
@@ Line 25: / Line 25: @@
 ==Satellite Orbit==
 **Part 1**\\
+{{ :repository:satellite_orbit.png?nolink&600|}}
 Copy and paste the following [[https://www.glowscript.org/#/user/nrosenmund/folder/Public/program/Newton'sLawOfGravitationandEnergyConservationSTUDENT | GlowScript code]] into your own GlowScript account. Read through the code and predict what might happen during the simulation
   - Run the program, observe and describe what happened and how it differed from your predictions
@@ Line 71: / Line 72: @@
 get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js')
 rom __future__ import division
-from visual import *                                        ######################
+from visual import *
-from visual.graph import *                                  ##                  ##
+from visual.graph import *
-                                                            ##  DO NOT CHANGE   ##
-#Window setup                                               ##  Window Setup    ##
+#Window setup
-scene.range = 7e7                                           ##       or         ##
+scene.range = 7e7
-scene.width = 1024                                          ##    Objects       ##
+scene.width = 1024
-scene.height = 760                                          ######################
+scene.height = 760
 #Objects
@@ Line 141: / Line 142: @@
 ===Handout===
 **Part 1** \\
+{{ :repository:satellite_orbit_answer_1.png?nolink&600 |}}
   - The satellite begins with a velocity pointing away from the Earth. Eventually, the satellite comes to a complete stop, then accelerates towards the Earth until it crashes.
   - Approximately 6 seconds
@@ Line 166: / Line 169: @@
     - Circular motion means that the satellite will remain the same distance away from the Earth throughout the entire orbit. Additionally, the magnitude of the satellite's velocity will also remain constant. Therefore, if the orbit of the satellite is circular, then its potential and kinetic energy will remain constant throughout, and checking the graphs will prove if you have achieved circular orbit
 **Part 2**\\
+  -Claim Support/Refutation:
+    -Claim: Energy is not conserved in the Earth-satellite system.
+      -Refute; no work is being done on the system. Instead there are internal energy transfers. Although the energy may transform from kinetic to potential, it will always sum to a constant amount, and the total mechanical energy is conserved
+    -Claim: The magnitude of the satellite's velocity is constant.
+      -Refute; this is only true for circular orbits (Part 1 Problem 13). As answered in Part 1 Problem 10c, the velocity increases towards the perigee and decreases towards the apogee
+    -Claim: Changing the mass of the Earth and/or the satellite changes the force of gravity involved in this system.
+      - Support; provided Newton's equation of gravitation ($F_{g}=\dfrac{Gm_{1}m_{2}}{r^2}$), it becomes clear that changing either of the masses will affect the magnitude of the force (although the direction will still point from one mass to the other)
+    - Claim: If the distance between the Earth and the satellite is doubled, the force between the Earth and the satellite is halved
+      - Refute; because the force is inversely proportional to the distance squared, a two-fold increase in distance results in the force being quartered
+        - $\dfrac{Gm_{1}m_{2}}{(2r)^2}=\dfrac{Gm_{1}m_{2}}{4r^2}=\dfrac{1}{4}F_{g}$
+    - Claim: Changing the initial momentum of the satellite changes the shape of the satellite's orbit around the Earth
+      - Support; mass of the satellite does not affect the orbit, while velocity does affect the shape. Therefore, altering the initial momentum, which is mass times velocity, will alter the size and shape of the orbit
+        - At velocities less than circular speed (Part 1 Problem 13b), the orbit is elliptical
+        - At velocities exactly at circular speed,  the orbit is circular
+        - At velocities greater than circular speed, but less than escape speed (defined as $v=\sqrt{\dfrac{2GM}{r}}$) the orbit is elliptical
+        - At velocities exactly at escape speed the orbit is parabolic
+        - At velocities greater than escape speed, the orbit is hyperbolic
+        - Larger satellite mass results in smaller orbit size
+  - Claims + Evidence
+    - Claim: The kinetic energy of the satellite increases as its distance to the Earth decreases
+      - Evidence: Based on Newton's Law of Gravitation, as the satellite gets closer to the Earth, the gravitational force acting on the satellite increases. Based on Newton's 2nd Law, force is directly proportional to acceleration, so the satellite will begin to accelerate, or in other words, increase its velocity. A larger velocity corresponds to a greater amount of kinetic energy, as proven by the equation for kinetic energy (Part 1 Problem 10d)
+    - Claim: The potential energy of the satellite increases as its distance from the Earth increases
+      - Evidence: When a satellite orbits a planet, it has negative potential energy and is bounded to the planet. Moving the satellite closer to Earth makes its potential energy become more negative. As the satellite drifts farther away from the Earth, its potential energy will approach zero (see Part 1 Problem 10d)
+    - Claim: The total mechanical energy of the satellite remains constant throughout the orbit of the satellite
+      - Evidence: If you were to calculate the kinetic and potential energy of the satellite at any point during its orbit, you will find that the sum will always be the same negative value (negative because it is bounded, as explained above). The total energy does not change because no work is being done on the system: only a transfer of internal energies. You can also see this in the graphs displayed below your code; the line or bar representing total mechanical energy will remain fixed
+    - Claim: The net force acting on the Earth and the satellite are equal and opposite
+        - Evidence: According to Newton's 3rd Law of Motion, every action has an equal and opposite reaction. That means that the gravitational force from the Earth acting on the satellite is equal in magnitude and opposite in direction to the force from the satellite acting on the Earth
+      - Claim: The mass of the satellite does not affect the orbital shape
+        - Evidence: We discovered earlier that the shape of the orbit is dependent on the speed of the satellite, not the mass (Part 2 Problem 1eI). You can test this by changing the values of "mSatellite" in your code (line 17), and you will find the shape of the orbit stays the same regardless
+      - Claim: The larger the radius between the satellite and the Earth, the smaller the velocity
+          - Evidence: This has already be thoroughly discussed in Part a of this problem
+  - Connecting code and real-life physics concepts
+    - Initial parameters/conditions and their effects
+      - The relationships between momentum, mass, G, orbital shape, net force, kinetic energy, potential energy, and total mechanical energy have been thoroughly explored and discussed in parts 1 and 2
+    - Origin of Earth-Moon system
+      - There are three main theories that explain how the Earth-Moon system originated:
+        - Early in the life of the Solar System as the planets were just beginning to form, the Earth and Moon could have formed together in a process known as accretion
+        - At some point in the history of the Earth, the moon could have originated elsewhere and it was captured by the gravity of the Earth. In order for this to have occured, the moon would have had to slow down to less than escape speed (Part 2 Problem 1eIC)
+        - By far the most popular theory, sometime long ago, a Mars-sized object (commonly called Theia) collided with the Earth, ejecting a large amount of material from Earth. This material then accreted to form what is now the Moon. This accounts for the axial tilt of the Earth.
+    - Satellite-Earth orbit
+      - The satellite is always falling towards the Earth; however, so long as it is travelling fast enough, the satellite essentially moves out of the way of the Earth as fast as it is falling. Basically, the effects of the satellite's velocity and the gravitational force from the Earth are balanced.
+===Code===
+[[https://www.glowscript.org/#/user/nrosenmund/folder/Public/program/Newton'sLawOfGravitationandEnergyConservationTEACHER | Link]]
+<code Python [enable_line_numbers="true", highlight_lines_extra="18,24,34,56,66,69"]>
+GlowScript 2.7 VPython
+get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js')
+rom __future__ import division
+from visual import *
+from visual.graph import *
+#Window setup
+scene.range = 7e7
+scene.width = 1024
+scene.height = 760
+#Objects
+Earth = sphere(pos=vector(0,0,0), radius=6.4e6, texture=textures.earth)
+Satellite = sphere(pos=vector(6.6*Earth.radius, 0,0), radius=1e6, color=color.orange, make_trail=True)
+#Parameters and Initial Conditions
+mSatellite = 1000
+pSatellite = vector(-1500*mSatellite,2598*mSatellite,0)
+G = 6.67e-11
+mEarth = 5.98e24
+r = (Earth.pos - Satellite.pos)
+g1 = gcurve(color=color.cyan,label="kinetic energy")
+g2 = gcurve(color=color.red,label="gravitational energy")
+g3 = gcurve(color=color.green,label="total mechanical energy")
+#Time and time step
+t = 0
+tf = 60*60*24*10
+dt = 1
+graphv = gdisplay(xmin=-0.25, xmax=1.25, ymin=-12e10, ymax=12e10, ytitle="Energy")
+g4 = gvbars(gdisplay = graphv, color = color.red, delta = 0.2, label = "Kinetic Energy")
+g5 = gvbars(gdisplay = graphv, color = color.blue, delta = 0.2, label = "Potential Energy")
+g6 = gvbars(gdisplay = graphv, color = color.green, delta = 0.2, label = "Total Energy")
+#MotionMap/Graph
+FSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=4000, labelMarkerOrder=False)
+pSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=0.2, markerColor=color.blue, labelMarkerOrder=False)
+#Calculation Loop
+ev = scene.waitfor('click')
+while t < tf:
+    rate(6000)
+    g4.delete()
+    g5.delete()
+    g6.delete()
+    Fnet = vector(0,0,0)
+    r = (Earth.pos - Satellite.pos)
+    Fnet = vector(G*mEarth*mSatellite/(mag(r)**2)*(r/mag(r)))
+    pSatellite = pSatellite + Fnet*dt
+    Satellite.pos = Satellite.pos + (pSatellite/mSatellite)*dt
+    if mag(Satellite.pos) < Earth.radius:
+        text(text='You Crashed!!', pos=vec(0, 4e7, 0), color = color.red, depth=1, height= 7e6)
+        break
+    FSatelliteMotionMap.update(t, Fnet)
+    pSatelliteMotionMap.update(t, pSatellite)
+    t = t + dt
+    KE = 1/2*mSatellite*mag(pSatellite/mSatellite)**2
+    PE = G*mSatellite*mEarth/mag(r)
+    g1.plot(t, KE)
+    g2.plot(t, PE)
+    g3.plot(t, KE+PE)
+    g4.plot(0, KE)
+    g5.plot(0.5, PE)
+    g6.plot(1.0, KE+PE)
+    #Earth Rotation (IGNORE)
+    theta = 7.29e-5*dt
+    Earth.rotate(angle=theta, axis=vector(0,0,1), origin=Earth.pos)</code>
+----
+====See Also===