Lab 2

Faculty Notes

Time is of the Essence

General Comments

• For students at the introductory level, the problem will need to be simplified by thinking of the distance being measured as if the world were flat. For students with more advanced mathematical backgrounds, the earth may be treated more realistically as a sphere. In this case, spherical coordinates can be used.
• It is assumed that the satellites are in geostationary orbit. It would be a challenge to have students explore satellites in one of the other types of orbit.
• It is possible that students will get different numbers for the distances needed depending on their source of information. Some web sites provide driving distances and not the straight-line distances.
• Students need to understand that telecommunications are sent at the speed of light. Sound from a Public Address speaker travels through air at the speed of sound.

Technology Problem
Simplified and Realistic Versions

The times required for a signal to travel between different locations may surprise the student. From their research, students should have discovered that geostationary satellites are located about 22,240 miles above the Earth. If this height is used, the distance the signal travels from Los Angeles to New Your via the satellite is approximately 45,000 miles. This is found using the Pythagorean Theorem. The time it takes for the signal to travel this distance using the speed of light is approximately 0.48 seconds. The angle of elevation of the signal from Los Angeles to the satellite is approximately 85°; the angle of depression from the satellite to New York City is approximately 88°. The time it takes for the signal to travel from New York City to Chicago via the satellite using the speed of light is approximately 0.24 seconds.

Additional Questions

For further exploration of the technology problem, you may wish to include the following questions for the simplified version of the technology problem.

• Assume the satellite normally used is unavailable due to other traffic. In that case, the communications are routed through another less powerful satellite, which is also over Chicago. This satellite needs to send the signal down to Chicago so it can be boosted and sent back up to the satellite and on to New York City. If the broadcast team is forced to use this less powerful satellite, how does the time delay change?
• Once the signal is completed in New York City, it is then sent to Denver, Colorado. This signal must also pass through the satellite over Chicago. How long does it take for the signal in the original question to go from New York City to Los Angeles, get a response, and return to New York City, then go to Denver?

Spinoffs

1. The Speaker Placement Problem (Speed of Sound) Rebecca is planning on attending a concert in an elliptical-shaped football stadium. She will be sitting in the end-zone bleachers directly in front of the scoreboard, which also contains the sound system. The length of the stadium is 550 feet. When the opening announcements are sent out she will hear them immediately, and a bit later she will hear an echo of the announcement as it returns after bouncing off the opposite wall. Rebecca's friend Beth will be sitting in the seats at the opposite end zone. The only speakers that exist for the stadium are located in the scoreboard.
   1. How long a delay is there before Beth hears the same announcement that Rebecca hears?
   2. How long does it take for Rebecca to hear the echo of the announcements?
   3. It has been decided that the volume needed to send the sound the entire length of the stadium would be so loud that it would endanger to the hearing of those fans sitting in the seats directly under the scoreboard. If the volume is reduced to a safe level, it will be necessary to position an additional speaker in the stadium so that the level will be loud enough for those in the opposite end zone to hear and enjoy the concert. The second speaker must be set with a delay so that when the sound from the original speaker reaches the additional speaker they are synchronized.
      1. If the second speaker were to be placed at the 50-yard line of the stadium, halfway between Beth and Rebecca, what time delay is needed so that the music stays synchronized?
      2. If the second speaker were to be placed at the 30-yard line of the stadium, what time delay is needed so that the music stays synchronized?
2. Comparison of Speed of Light and Speed of Sound
   
   Paul and Ray are at the playoff game in Los Angeles. The game announcer and the Public Address (PA) speakers are 90 yards away from Ray and Paul's seats. How long does it take for them to hear the announcements made at the stadium?
   
   In the ninth inning, the Cubs hold a 10-2 lead. Paul and Ray decide to leave early to beat the traffic. They leave the game and walk to their car while the game is still in progress. Paul is wearing a Walkman television and continues to listen to a local broadcast of the game. Ray relies on the crowd noise for his information. The Los Angeles Dodgers score a run. Who do you think will hear the roar of the crowd first, Ray or Paul?
   
   Estimate how far Ray and Paul are from the stadium if one of them hears the roar of the crowd one second before the other. Now calculate the exact distance.

Bibliotechnology Research

Although the web is a useful tool for research, some of the facts necessary to complete this lab might be more easily found if the student uses printed sources. You might also consider having them research the Global Positioning System and how it ties in with this project.

Sample web sites for Research

• For distances
  http://www.indo.com/distance (the best — it gives "as the crow flies" distances)
  http://www.MapQuest.com
  http://www.earthmaps.com (http://www.DeLorme.com)
  
• For telecommunications information
  http://www.howstuffworks.com (search for satellites)
  http://www.celestrak.com/columns (information on the Clarke Belt)
  http://www.encarta.msn.com (communications satellite)
  http://octopus.gma.org/surfing (remote sensing, satellite imagery)
  http://www.satobs.org/satintro.html (satellite observation)

Mathematics Tools

The students will need to be familiar with the following topics. It may be necessary to do some 'mini-lessons' on some of the topics.

• Pythagorean Theorem
• Distance = Rate * Time
• Right Triangle Trigonometry
• Equation solving using rational expressions
• Dimensional Analysis
• Precision and Accuracy
• Spherical Coordinates

Thesis Defense

The student presentation will include:

1. a thesis statement regarding the placement of the satellite;
2. a scale model of the basic problem;
3. the mathematical solution to the problem including all calculations;
4. the basic algorithm for a graphing calculator or computer program(s) to calculate the distance, time, and angle of elevation; and
5. the results of the background research on satellites.