Calculating a Spy Satellite's Orbit Parameters

In summary, the spy satellite in circular orbit around Earth makes one revolution in 6.00 hours. To find the satellite's height above Earth's surface, we can use the equation GMm/r^2 = m(v^2/r) and cancel out the mass of the satellite. We can then find another equation in terms of v and r using the 6.00 hour period, and solve for r and v using the two equations and two unknowns. This will give us the height and acceleration of the satellite.
  • #1
Jtappan
96
0

Homework Statement



A spy satellite is in circular orbit around Earth. It makes one revolution in 6.00 hours.

(a) How high above Earth's surface is the satellite?
________ km
(b) What is the satellite's acceleration?
________ m/s2

Homework Equations



GMm/r^2 = m(v^2/r)


The Attempt at a Solution



I don't understand how you find these answers if you don't have the mass of the satelite.
 
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  • #2
masses cancel...

GMm/r^2 = m(v^2/r)

you can cancel m from both sides

GM/r^2 = v^2/r

you need to look up M the mass of the earth...

Get another equation in terms of v and r (use the 6.00hr)

Then you can solve for r and v, since you have 2 equations with 2 unknowns.
 
  • #3
Additionally, without knowing the specific values for the radius of the Earth and the orbital period, it is difficult to provide an accurate answer. However, we can use the formula GMm/r^2 = m(v^2/r) to calculate the altitude of the satellite and its acceleration.

(a) To find the altitude, we can rearrange the formula to solve for r (altitude). We can assume the mass of the satellite (m) is negligible compared to the mass of the Earth (M), so the formula becomes GM/r^2 = v^2/r. We can substitute the values for the radius of the Earth (6,371 km) and the orbital period (6 hours or 21,600 seconds) to get:

GM/r^2 = (2πr/21,600)^2
GM/r^2 = 4π^2r^2/466,560,000

We can then solve for r by multiplying both sides by r^2 and rearranging the equation:

GM = 4π^2r^3/466,560,000
r^3 = 466,560,000GM/4π^2
r = (466,560,000GM/4π^2)^1/3

Plugging in the values for the gravitational constant (G = 6.67 x 10^-11 Nm^2/kg^2) and the mass of the Earth (M = 5.97 x 10^24 kg), we get:

r = (466,560,000 x 6.67 x 10^-11 x 5.97 x 10^24 / 4π^2)^1/3
r = (1.11 x 10^14)^1/3
r = 6.69 x 10^6 km

Therefore, the satellite is approximately 6,690 km above the Earth's surface.

(b) To find the acceleration, we can use the formula GMm/r^2 = m(v^2/r) and solve for the acceleration (a). Again, we can assume the mass of the satellite is negligible compared to the mass of the Earth, so the formula becomes GM/r^2 = v^2/r. We can substitute the values for the radius of the Earth (6,371 km) and the orbital period (6 hours or 21,600 seconds) to get:

GM/r^2 = (
 

FAQ: Calculating a Spy Satellite's Orbit Parameters

How is a spy satellite's orbit calculated?

A spy satellite's orbit is calculated using a combination of mathematical equations, data from ground stations, and observations of the satellite's movements. This process is known as orbital mechanics.

What factors are taken into account when calculating a spy satellite's orbit parameters?

When calculating a spy satellite's orbit parameters, factors such as the satellite's mass, altitude, speed, and the gravitational pull of the Earth are taken into account. Other factors, such as atmospheric drag and solar radiation pressure, may also be considered.

Can a spy satellite's orbit parameters change over time?

Yes, a spy satellite's orbit parameters can change over time due to various factors such as atmospheric drag, gravitational forces from other celestial bodies, or intentional adjustments made by ground control. These changes must be continually monitored and calculated to ensure the satellite remains in its desired orbit.

How accurate are calculations of a spy satellite's orbit parameters?

The accuracy of calculations for a spy satellite's orbit parameters can vary depending on the available data and the complexity of the orbit. Typically, these calculations have a margin of error of a few kilometers, which is acceptable for the purposes of tracking and controlling the satellite's movements.

How do scientists use a spy satellite's orbit parameters in their research?

Scientists use a spy satellite's orbit parameters to track and monitor the satellite's movements and to predict its future path. They may also use this data to study the Earth's atmosphere, weather patterns, and other phenomena from a unique perspective. Additionally, these parameters can be used to plan and execute maneuvers to adjust the satellite's orbit as needed.

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