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Mathematics Standards
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Algebra |
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Computation and Estimation |
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Conceptual Underpinnings of Calculus |
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Discrete Mathematics |
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Functions |
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Mathematical Connections |
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Mathematics as Problem Solving |
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Mathematics as Reasoning |
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Number and Number Relationships |
Science Standards
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Physical Science |
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Science and Technology |
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Science as Inquiry |
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Unifying Concepts and Processes |
Questions for Discussion
- How does the Shuttle stay in orbit? Use the following two equations
that describe the force acting on an object. The first equation represents
the force of gravity acting on the Shuttle.
Where:
F 1 = Force of gravity acting on the Shuttle
G = Universal gravitational constant
m e = Mass of Earth
m s = Mass of the Shuttle
r = Distance from center of Earth to the Shuttle
The second equation represents the force acting on the Shuttle that
causes a centripetal acceleration,
This is an expression of Newton's second law, F=ma.
F 2 = Force acting on the Shuttle that causes uniform circular
motion (with centripetal acceleration)
v = Velocity of the Shuttle
These two forces are equal: F1 =F 2
In order to stay in a circular orbit at a given distance from the center
of Earth, r, the Shuttle must travel at a precise velocity, v.
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- How does the Shuttle change its altitude?
From a detailed equation relating the Shuttle velocity with the Shuttle
altitude, one can obtain the following simple relationship for a circular
orbit. Certain simplifying assumptions are made in developing this equation:
1) the radius of the Shuttle orbit is nearly the same as the radius
of Earth, and 2) the total energy of the Shuttle in orbit is due to
its kinetic energy, 1/2 mv2; the change in potential energy
associated with the launch is neglected.
= orbital period. The time it takes the Shuttle to complete one revolution
around Earth
 =
the change in Shuttle velocity
 =
the change in Shuttle altitude
For example:
Consider a Shuttle in a circular orbit at 160 nautical miles (296.3
km) altitude. Determine the new altitude caused by the Shuttle firing
a thruster that increases its velocity by 1 m/s. First, calculate the
orbital period, , from the above equation.
Next, use the period and the applied velocity change to calculate the
altitude change.
This altitude change is actually seen on the opposite side of the orbit.
In order to make the orbit circular at the new altitude, the Shuttle
needs to apply the same at the other side
of the orbit.
In the discussion and example just given, we state that the equations
given are simple approximations of more complex relationships between
Shuttle velocity and altitude. The more complex equations are used by
the Shuttle guidance and navigation teams who track the Shuttles' flights.
But the equations given here can be used for quick approximations of
the types of thruster firings needed to achieve certain altitude changes.
This is helpful when an experiment team may want to request an altitude
change. Engineers supporting the experiment teams can determine approximately
how much propellant would be required for such an altitude change and
whether enough would be left for the required de-orbit burns. In this
way, the engineers and experiment teams can see if their request is
realistic and if it has any possibility of being implemented.
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