Using the background knowledge presented in this lesson, students will
calculate two components of satellites, namely altitude and velocity.
This lesson is comprised of two main problems. In each problem,
students work in groups and use algebraic thinking to determine
solutions to the problems posed. Suggested methods for solving these
problems are described in detail. The teacher is encouraged to
highlight alternative methods for solving the problems as discovered by
students, when appropriate.
Weather satellites are important for collecting data about Earth.
These unmanned spacecraft carry a variety of sensory equipment that
scans Earth and electronically communicates the data back to scientists
on Earth. Two types of environmental satellite systems named for their
orbital characteristics, geostationary and polar-orbiting satellites,
provide the data for scientists to study our environment.
Geostationary satellites orbit at a speed that enables them to
remain constantly over the same area of Earth. This position provides
constant "viewing" of a specific area of Earth. The Geostationary
Operational Environmental Satellites are an important system for the
United States. These satellites are maneuvered into orbits that ensure
constant coverage of the Western Hemisphere. They continually produce
infrared images, weather charts, ice charts, and other data important
to the study of the environment.
Polar-orbiting satellites operate in orbits that cause them to
pass almost over the poles of the Earth. Many of these satellites
operate in Sun-synchronous orbits. The Sun-synchronous orbits are
designed so that the satellite passes over the same terrain at the same
time each day. This orbital feature is important for the collection of
data about Earth.
To simplify the notion of orbits for middle school students, the
orbit discussed in these activities are assumed to be circular. The
orbital period is the time required to complete one orbit around Earth.
Engage the class in a large-group discussion about the satellite
images they see on local television news broadcasts. In the discussion,
help the students distinguish among animations of Shuttle missions,
photographs from the Space Shuttle, pictures from remote robotic
explorations such as Pathfinder on Mars, Hubble Space Telescope images,
and satellite photographs of Earth. If possible, obtain a videotape of
a television weather report of a fast-moving storm that has been
tracked and shown in nearly real time. If the videotape cannot be
obtained, ask students to recall watching such events on television.
(Alternatively, students can use Internet resources to find video
As part of the discussion, ask students to speculate how the
television station obtained these images. Some students may know that
some TV weather graphics are made with satellite data.
This discussion can lead to an opportunity to present the
notions of geostationary and polar-orbiting satellites. The notion of
polar-orbiting satellites (those whose orbits take them above both
poles) may be easier for students to understand. They seem to
understand the notion of the ground track of polar-orbiting satellites.
The reason may be that the movement of these satellites is similar to
that of an airplane in the sky. Thus, presenting the polar-orbiting
notion first may be desirable. The class discussion will determine if
this seems appropriate.
Note: For computations in this lesson, we use 3.14 as the value of pi.
Pose the following question to the class: How do you determine the
velocity of a polar-orbiting satellite when the altitude and the period
of the orbit are known? For example, if a 750-mile high satellite
orbits Earth every 3 hours, 45 minutes, how fast is it traveling? That
is, what is the velocity of the satellite?
Allow the students to work in small groups to draw pictures of
the situation and attempt to think through the solution to the problem
without doing the actual computation. Encourage them to think about the
information they already know, and what information they need to solve
the problem. Students can record this information on their papers.
Once the group has a plan to compute the velocity of the satellite, have each student in the group determine the velocity.
Sample method for solving the example:
Step 1: Change 3 hours, 45 minutes to:
3 45/60 = 3¾ = 3.75 hours
Note that a fraction calculator can be used for this conversion.
The radius of an orbit is calculated from the center of the
Earth. Consequently, the radius of an orbit equals the radius of Earth
plus the altitude of the satellite above the Earth.
Step 2: According to the formula C = 2 × pi, the distance traveled is:
C = 2 × pi × (3960.5 + 750) ≈ 29,581 miles
Note: 3,960.5 miles is the radius of the Earth.
Step 3: The distance of the orbit (29,581 miles) is traveled each 3.75 hours. Therefore, the speed—or velocity—of the satellite is determined by dividing distance by time:
29,581 miles ÷ 3.75 hours ≈ 7,888 miles per hour
|Altitude above Earth ||Period ||Velocity |
|170 miles ||1.5 hours ||~17,293 mph |
|20,150 miles ||20 hours ||~7,570 mph |
When students are sharing solutions,
emphasize solutions' methods. Talk about the different approaches (if any) the students used to reach the same solution.
Pose the following question to the class: How do you find the
altitude of a geosynchronous satellite that is orbiting Earth above the
equator? The velocity of the satellite must be known. Using a value for
the velocity, we could ask, "If there is a satellite traveling at
6,900 miles per hour, how high must it be orbiting?"
This question is sometimes difficult for students because
there is only one number in the problem. Putting students into groups
to draw pictures and think about what is known about the problem will
usually help them identify all the pertinent data. Drawing pictures
will help their discussions.
One possible solution method:
Step 1: A geostationary satellite "orbits" the Earth once
every 24 hours. The time of the orbit is what makes it appear not to
move. It is moving in unison with Earth's rotation.
Step 2: If a satellite travels 6,900 miles in 1 hour, how far does it travel in the 1-day orbit (a period of one day)?
6,900 miles per hour × 24 hours = 165,600 miles
Step 3: 165,600 miles is the circumference (C) of a circle, and C = 2 × pi r. Therefore,
pi × r = 165,600 ÷ 2 = 82,800 miles
r = 82,800 ÷ pi ≈ 26,369 miles.
Step 4: 26,369 miles is the radius of a circle whose center
is the center of the Earth. To determine the satellite's altitude above
the surface of the Earth, the radius of Earth (3,960.5 miles) must be
subtracted from the radius of the circle. Therefore, the altitude above
the Earth is:
26,369 - 3,960.5 = 22,408.5 miles
Once again, it is important to have the students share their
solutions to this problem. They may see different methods for solving
this problem. When students share their thinking, some of their
classmates will gain new or heightened understanding of the lesson
Adapted from Orbiting Satellites in Mission Mathematics, Linking Aerospace and the NCTM Standards, a NASA/NCTM project, NCTM 1997. Images from Web sites are credited by inclusion of the URL and/or direct links to their source.