Question

A 1 046-kg satellite orbits the Earth at a constant altitude of 101-km. (a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 194 km? How is the total energy of an object in circular orbit related to the potential energy? MJ (b) What is the change in the system's kinetic energy? Is the satellite moving faster or slower when it's orbit is at a higher altitude? MJ (c) What is the change in the system's potential energy? You appear to have correctly calculated the change in potential energy using your incorrect result from parts (a) and (b). MJ

Answer 1

To move the satellite into a circular orbit with a higher altitude, we need to calculate the change in potential energy. The total energy of an object in a circular orbit is the sum of its kinetic and potential energy. The change in kinetic energy is the difference in total energies of the two orbits. The change in potential energy and kinetic energy can be calculated using the equations provided.

Step-by-step explanation:

To move the satellite into a circular orbit with altitude 194 km, we need to calculate the change in potential energy. The potential energy of an object in a circular orbit is given by the equation PE = - GMm/r, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance between the center of the Earth and the satellite. The difference in potential energy between the two orbits is then the change in potential energy. This can be calculated as ΔPE = PE2 - PE1, where PE2 is the potential energy at altitude 194 km and PE1 is the potential energy at altitude 101 km. The change in potential energy can be converted to energy using the equation ΔPE = ΔE, where ΔE is the energy added to the system.

To calculate the change in potential energy, we need to know the radius of the Earth at the two altitudes. The radius of the Earth is approximately 6,371 km. The radius at altitude 101 km is given by r1 = 6,371 km + 101 km. The radius at altitude 194 km is given by r2 = 6,371 km + 194 km.

Using these values, we can calculate the change in potential energy:

ΔPE = (- GMm/r2) - (- GMm/r1)

Next, we need to calculate the change in kinetic energy. The total energy of an object in a circular orbit is the sum of its kinetic and potential energy. The change in kinetic energy is then the difference in the total energies of the two orbits. This can be calculated as ΔKE = TE2 - TE1, where TE2 is the total energy at altitude 194 km and TE1 is the total energy at altitude 101 km. Since the satellite is in a circular orbit, its kinetic energy is given by KE = -GMm/2r. Using this equation, we can calculate the total energies at the two altitudes:

TE = PE + KE

Finally, we can calculate the change in potential energy and change in kinetic energy using the given masses of the satellite and the equations above.

1. (a) The energy added to the system to move the satellite into a circular orbit with altitude 194 km is ΔE = ΔPE = - GMm/r2 - (- GMm/r1).
2. (b) The change in the system's kinetic energy is ΔKE = TE2 - TE1 - ΔPE. Since the satellite is moving into a higher altitude, the change in kinetic energy will be negative, indicating that the satellite is moving slower in the higher orbit.
3. (c) The change in the system's potential energy is given by ΔPE = - GMm/r2 - (- GMm/r1).