As a research scientist for your government, you are in a meeting in which a dif

Question

As a research scientist for your government, you are in a meeting in which a difficult situation has arisen. According to the director of your agency, your country needs a spy satellite to gather intelligence on terrorists operating in Pontianak, Indonesia (at longitude 112.7o) and Kampala, Uganda (at longitude 32.5o). Both cities reside close to the equator; the arc angle between the two cities is simply the difference in their longitudes. Normally, a country would send up two satellites to maintain geosynchronous* orbits above each city. But only one satellite is available for launch so it will have to pass over one city to the next in its orbit. Complicating matters is the need to synchronize the data collected over the two cities. Your agency demands that there can be no more than 5 hours of travel between the time the satellite passes over one city to the next. At what maximum altitude will your agency need to position the satellite to satisfy these requirements? The mass of the satellite is 1,240 kg. For other physical constants, please refer to your textbook or perform a simple Web search. A top view of the Earth (that is, looking down on the North Pole) is pictured above.
Important to the mission is to maintain a slow enough pass over each city to allow the cameras time to take large numbers of pictures. Given that the camera on the satellite always points towards the center of the Earth, at what speed does the camera's field of view track across the surface of the Earth?

As a research scientist for your government, you are in a meeting in which a difficult situation has arisen. According to the director of your agency, your country needs a spy satellite to gather intelligence on terrorists operating in Pontianak, Indonesia (at longitude 112.7o) and Kampala, Uganda (at longitude 32.5o). Both cities reside close to the equator; the arc angle between the two cities is simply the difference in their longitudes. Normally, a country would send up two satellites to maintain geosynchronous* orbits above each city. But only one satellite is available for launch so it will have to pass over one city to the next in its orbit. Complicating matters is the need to synchronize the data collected over the two cities. Your agency demands that there can be no more than 5 hours of travel between the time the satellite passes over one city to the next. At what maximum altitude will your agency need to position the satellite to satisfy these requirements? The mass of the satellite is 1,240 kg. For other physical constants, please refer to your textbook or perform a simple Web search. A top view of the Earth (that is, looking down on the North Pole) is pictured above.
Important to the mission is to maintain a slow enough pass over each city to allow the cameras time to take large numbers of pictures. Given that the camera on the satellite always points towards the center of the Earth, at what speed does the camera's field of view track across the surface of the Earth?

Explanation / Answer

For the given condition, the satellite must travel opposite to the direction of rotation of earth.

So, for minimum height case, the satellite should travel from Uganda to Indonesia in opposite direction.

In 5 hrs, angle covered by earth = 360*5/24 = 75 degrees

Angle to be covered by the satellite = 360-75-(112.7-32.5)

= 204.8 degrees.

So, angular velocity = 1.98 *10^-4 rad/s

For satellite, angular velocity = sqrt(G*Me/r^3)

r = 21620070 m

= 21620 km

Altitude from earth surface = r - Re

= 15249 km

Speed of satelllite = r*angular velocity = 4280 m/s

Speed of cities = Rearth * angular vel of earth = 463.3 m/s

So, relative speed = 4280 + 463 = 4743 m/s

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