• Use mathematical representations of Newton's Law of Gravitation and Coulomb's Law to describe and predict the gravitational and electrostatic forces between objects HS-PS2-4
• Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy and associated with the relative positions HS-PS3-2

• Kinematics
• Conservation of Energy
• Newton's 2nd Law
  • Net Force
• Momentum
• Newton's Law of Gravitation
  • $F_{g}=\dfrac{Gm_{1}m_{2}}{r^2}$

• Interpret code and relate to physics concepts
• Copy and paste code
  • Replicate motion maps
• Add variables to premade graph

Part 1
Copy and paste the following GlowScript code into your own GlowScript account. Read through the code and predict what might happen during the simulation

1. Run the program, observe and describe what happened and how it differed from your predictions
2. How long does the simulation run for?
3. What is the initial momentum of the satellite? Which direction?
4. Which line of code indicates Newton's Law of Gravitation?
5. How far way is the satellite from the Earth initially?
6. What is the mass of the satellite? What is the mass of the Earth?
7. What does the following line of code mean in physics terms? What does it do?
   1. “pSatellite = pSatellite + Fnet*dt” (line 48)
8. How is velocity represented in the code?
9. Modify the code in order to make the satellite orbit the Earth. What changes did you make?
10. Qualitatively and quantitatively described the orbit of your satellite in terms of the following, and provide a sketch:
    1. Shape:
    2. Net Force:
    3. Velocity:
    4. Kinetic Energy:
    5. Potential Energy:
    6. Total Energy:
11. Add a net force vector to your satellite
12. Add a bar graph for the total mechanical energy
13. Challenge: modify the code to make the orbit of the satellite circular. What evidence do you have that it is circular (besides visual inspection)?

Part 2
Conceptual Physics Claim, Evidence, and Reasoning (C.E.R.):

1. Do you support or refute the following claims? Provide evidence and reasoning:
   1. Energy is not conserved in the Earth-satellite system.
   2. The magnitude of the satellite's velocity is constant.
   3. Changing the mass of the Earth and/or the satellite changes the force of gravity involved in this system.
   4. If the distance between the Earth and the satellite is doubled, the force between the Earth and the satellite is halved.
   5. Changing the initial momentum of the satellite changes the shape of the satellite's orbit around the Earth.
2. For the following concepts, create a claim, cite your evidence, and provide your reasoning for each prompt:
   1. The kinetic energy of the satellite.
   2. The potential energy of the satellite.
   3. The total mechanical energy of the system.
   4. Relationship between the net force on the Earth and net force on satellite.
   5. Relationship between the mass of the satellite and the shape of the orbit.
   6. Relationship between radius and velocity of the satellite.
3. Respond to the following statements/questions using your prior knowledge and physics concepts
   1. Discuss how changing the parameters and initial conditions (initial momentum, mass, G) affect orbit shape, net force, kinetic energy, potential energy, total mechanical energy, and momentum.
   2. Using this simulation and your own experience, how can you explain how the Earth-Moon system could have originated?
   3. Why does the satellite remain in orbit without crashing to Earth?

Link

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 *                                  ##                  ##
                                                            ##  DO NOT CHANGE   ##
#Window setup                                               ##  Window Setup    ##
scene.range = 7e7                                           ##       or         ##
scene.width = 1024                                          ##    Objects       ##
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 = 15000
pSatellite = vector(30000000,0,0)
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")
 
 
#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")
g3 = gvbars(gdisplay = graphv, color = color.red, delta = 0.2, label = "Kinetic Energy")
g4 = gvbars(gdisplay = graphv, color = color.blue, delta = 0.2, label = "Potential Energy")
 
 
#MotionMap/Graph
pSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=0.2, markerColor=color.blue, labelMarkerOrder=False)
 
#Calculation Loop
ev = scene.waitfor('click')
while t < tf:
    rate(6000)
    g3.delete()
    g4.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
    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(0, KE)
    g4.plot(0.5, PE)
 
 
    #Earth Rotation (IGNORE)
    theta = 7.29e-5*dt
    Earth.rotate(angle=theta, axis=vector(0,0,1), origin=Earth.pos) 

—-

Part 1

1. 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.
2. Approximately 6 seconds
3. 3,0000,000 kg·m/s in the positive x-direction (line 18)
4. Line 50: “Fnet = vector(G*mEarth*mSatellite/(mag®**2)*(r/mag®))”
5. Line 14 indicates that the satellite is a distance of 6.6 times the Earth's radius away. The Earth's radius (line 13) is $6.4*10^6$ meters, therefore the satellite is initially $6.6*6.4*10^6=4.2*10^7$ meters away
6. The mass of the satellite is 15000 kg (line 17) and the mass of the Earth is $5.98*10^24$ kg (line 20)
7. The line of code that says “pSatellite = pSatellite + Fnet*dt” is located within the while loop and it updates the momentum of the satellite. In physics terms, this reads as “The updated momentum of the satellite is equal to the original momentum plus the impulse (force multiplied by time) acting on the satellite due to the force of gravity”
8. In line 49, the velocity is represented as “(pSatellite/mSatellite).” This comes from the fact that momentum is mass times velocity ($p=mv$), so dividing the momentum of the satellite by its mass, the velocity can be calculated
9. See updated code below
10. Satellite orbit description:
    1. Shape: The orbit is elliptical about the Earth
    2. Net Force: The strength of the gravitational force is strongest when the satellite is closest to Earth (perigee) and weakest when when it is furthest (apogee)
    3. Velocity: Similarly to net force, the velocity of the satellite is highest at its perigee and lowest at its apogee
    4. Kinetic Energy: Kinetic Energy is defined as $KE=\dfrac{1}{2}mv^2$ (line 56). Since kinetic energy is proportional to velocity squared, it makes sense that the kinetic energy is highest when the velocity is at a maximum (when the satellite is closest to Earth) and lowest when the velocity is at a minimum (when the satellite is furthest from Earth)
    5. Potential Energy: Gravitational potential energy is defined as $PE=-\dfrac{GMm}{r}$ (line 58). Since the gravitational constant and masses of the Earth and satellite remain fixed, the potential energy only varies as the distance between the Earth and satellite change. Given an inversely proportional relationship between potential energy and distance, the magnitude of the potential energy is greatest when the satellite is closest to Earth and smallest when the satellite is furthest from Earth
    6. Total Energy: The total mechanical energy in the Earth-satellite system is simply the sum of the kinetic and gravitational potential energies. This value is constant throughout the entire orbit, and always negative, which means that the satellite is bounded to Earth
11. See lines 38 and 56 below
12. See lines 34, 37, and 69 below
13. If the orbit of the satellite is circular, then it can be modeled using centripetal motion. Specifically, if the centripetal “force” is equal to the gravitational force, then we can find the initial velocity required to keep the satellite in circular orbit:
    1. $\dfrac{mv^2}{r}=\dfrac{GMm}{r^2}$
    2. Solving for velocity: $v=\sqrt{\dfrac{GM}{r}}$
    3. Plugging in known values: $v=\sqrt{\dfrac{6.67*10^-11{r}}$