Two Earth satellites, A and B, each of mass m, are to be launched into circular orbits about Earth's center. Satellite A is to orbit at an altitude of 7460 km. Satellite B is to orbit at an altitude of 20500 km. The radius of Earth REis 6370 km. (a) What is the ratio of the potential energy of satellite B to that of satellite A, in orbit? (b) What is the ratio of the kinetic energy of satellite B to that of satellite A, in orbit? (c) Which satellite (answer A or B) has the greater total energy if each has a mass of 11.4 kg? (d) By how much?

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To find the answers to these questions, we will first need to calculate the potential energy and kinetic energy of each satellite. Let's start by understanding the formulas for potential energy and kinetic energy in the context of satellite motion.

The potential energy (PE) of an object in orbit is determined by its altitude and mass. The formula for potential energy is:

PE = mgh

Where m is the mass of the object, g is the acceleration due to gravity, and h is the altitude or height from the surface of the Earth.

The kinetic energy (KE) of an object in orbit is determined by its mass and velocity. The formula for kinetic energy is:

KE = (1/2)mv^2

Where m is the mass of the object, and v is its velocity.

Now, let's solve each part of the question step by step.

(a) What is the ratio of the potential energy of satellite B to that of satellite A, in orbit?

To find the ratio of potential energies, we need to calculate the potential energy for each satellite.

For satellite A:
The altitude of satellite A (hA) = 7460 km + 6370 km (Earth's radius)
hA = 13830 km = 1.383 x 10^7 m

Potential energy of satellite A (PEA) = mghA = m x 9.8 m/s^2 x hA

For satellite B:
The altitude of satellite B (hB) = 20500 km + 6370 km (Earth's radius)
hB = 26870 km = 2.687 x 10^7 m

Potential energy of satellite B (PEB) = mghB = m x 9.8 m/s^2 x hB

The ratio of potential energy of satellite B to that of satellite A is given by:

PEB/PEA = (m x 9.8 m/s^2 x hB) / (m x 9.8 m/s^2 x hA)
= hB / hA

So, the ratio is hB / hA = 2.687 x 10^7 m / 1.383 x 10^7 m.

(b) What is the ratio of the kinetic energy of satellite B to that of satellite A, in orbit?

To find the ratio of kinetic energies, we need to calculate the kinetic energy for each satellite.

For both satellites A and B, they are in circular orbits, so their velocities (v) can be determined using the equation:

v = (GM/R)^0.5

Where G is the gravitational constant (6.67 x 10^-11 Nm^2/kg^2), M is the mass of the Earth (5.97 x 10^24 kg), and R is the distance from the satellite to the center of the Earth (Earth's radius + altitude).

For satellite A:
R = 6370 km + 7460 km = 13830 km = 1.383 x 10^7 m

vA = (GM/R)^0.5

For satellite B:
R = 6370 km + 20500 km = 26870 km = 2.687 x 10^7 m

vB = (GM/R)^0.5

The ratio of kinetic energy of satellite B to that of satellite A is given by:

KEB/KEA = [(1/2)m(vB^2)] / [(1/2)m(vA^2)]
= (vB^2) / (vA^2)

So, the ratio is (vB^2) / (vA^2) = [(GM/RB)^0.5 / (GM/RA)^0.5]^2 = (RB/RA)^2.

(c) Which satellite (answer A or B) has the greater total energy if each has a mass of 11.4 kg?

The total energy of a satellite in orbit is the sum of its potential energy and kinetic energy.

For satellite A:
Total energy of satellite A (TEA) = PEA + KEA

For satellite B:
Total energy of satellite B (TEB) = PEB + KEB

We can compare TEA and TEB to determine which satellite has the greater total energy.

(d) By how much?

To find out by how much the total energy differs, we need to calculate the absolute difference between TEA and TEB.

TEA - TEB = (PEA + KEA) - (PEB + KEB)

By evaluating this expression, we can determine the numerical difference between the total energies of satellite A and B.