A 7410-kg satellite has an elliptical orbit, as in the figure. The point on the orbit that is farthest from the earth is called the apogee and is at the far right side of the drawing. The point on the orbit that is closest to the earth is called the perigee and is at the far left side of the drawing. Suppose that the speed of the satellite is 2840 m/s at the apogee and 8050 m/s at the perigee.
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To better understand the orbit of the satellite, we can use the principle of conservation of mechanical energy. This principle states that the sum of the kinetic energy and potential energy of an object remains constant, as long as there are no external forces acting on it. In this case, the only significant force acting on the satellite is the gravitational force from the Earth.
The kinetic energy of an object can be calculated using the formula:
K.E. = (1/2) * mass * velocity^2
The potential energy of an object in a gravitational field can be calculated using the formula:
P.E. = mass * gravity * height
Where:
- mass is the mass of the satellite (7410 kg in this case)
- velocity is the speed of the satellite at a particular point in its orbit
- gravity is the acceleration due to gravity on Earth (approximately 9.8 m/s^2)
- height is the distance between the satellite and the Earth's surface at that point in its orbit
At the apogee, the speed of the satellite is given as 2840 m/s. We can calculate the potential energy at this point by considering the distance from the Earth's surface to the satellite at apogee. Let's assume this distance is represented by r.
At the perigee, the speed of the satellite is given as 8050 m/s. Similarly, we can calculate the potential energy at this point by considering the distance from the Earth's surface to the satellite at perigee, which we'll assume is represented by R.
Since the kinetic and potential energy are conserved throughout the orbit, we can equate them at apogee and perigee:
K.E. at apogee + P.E. at apogee = K.E. at perigee + P.E. at perigee
This equation allows us to solve for the distance from the Earth's surface at apogee (r) and perigee (R). The values of r and R will help us determine the shape and size of the satellite's elliptical orbit.
To solve the equation, we need additional information, such as the value of gravity at that point, or the heights above the Earth's surface at apogee and perigee.