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Chapter 3: Problem 164

A satellite is put into an elliptical orbit around the earth and has a velocity \(v_{P}\) at the perigee position \(P .\) Determine the expression for the velocity \(v_{A}\) at the apogee position \(A .\) The radii to \(A\) and \(P\) are, respectively, \(r_{A}\) and \(r_{P} .\) Note that the total energy remains constant.

Short Answer

Expert verified
\(v_A = \sqrt{v_P^2 + 2GM \left(\frac{1}{r_A} - \frac{1}{r_P}\right)}\)

Step by step solution

01

Apply Conservation of Mechanical Energy

The total mechanical energy for the satellite in its orbit is constant and can be expressed as sum of kinetic and potential energy at perigee (point P) equal to that at apogee (point A). The formula for mechanical energy is given by:\[ E = rac{1}{2}mv^2 - rac{GMm}{r} \]Where \(m\) is the mass of the satellite, \(v\) is the velocity, \(G\) is the gravitational constant, \(M\) is the mass of Earth, and \(r\) is the distance to the center of Earth.
02

Write the Energy Expressions for Perigee and Apogee

At perigee position P, the energy is:\[ E_P = \frac{1}{2}mv_P^2 - \frac{GMm}{r_P} \]At apogee position A, the energy is:\[ E_A = \frac{1}{2}mv_A^2 - \frac{GMm}{r_A} \]
03

Set the Energies Equal to Each Other

Since energy is conserved, equate the expressions for energy at perigee and apogee:\[ \frac{1}{2}mv_P^2 - \frac{GMm}{r_P} = \frac{1}{2}mv_A^2 - \frac{GMm}{r_A} \]
04

Simplify and Solve for v_A

Cancel out the mass \(m\) of the satellite from the equation (assuming \(m eq 0\)). Rearrange to isolate \(v_A\):\[ \frac{1}{2}v_P^2 - \frac{GM}{r_P} = \frac{1}{2}v_A^2 - \frac{GM}{r_A} \]\[ \frac{1}{2}v_A^2 = \frac{1}{2}v_P^2 + GM \left( \frac{1}{r_A} - \frac{1}{r_P} \right) \]\[ v_A^2 = v_P^2 + 2GM \left( \frac{1}{r_A} - \frac{1}{r_P} \right) \]\[ v_A = \sqrt{v_P^2 + 2GM \left( \frac{1}{r_A} - \frac{1}{r_P} \right)} \]
05

Final Expression for v_A

We have derived the expression for the velocity at apogee, \(v_A\):\[ v_A = \sqrt{v_P^2 + 2GM \left( \frac{1}{r_A} - \frac{1}{r_P} \right)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Satellite Orbital Dynamics
Satellite orbital dynamics is the study of how satellites move in their orbits around celestial bodies due to gravitational forces. These dynamics are governed primarily by Kepler's laws of planetary motion and Newton's law of universal gravitation. Satellites in orbit are influenced by the gravitational pull of Earth, keeping them in a specific path.
  • Gravity acts as a centripetal force, holding the satellite in orbit.
  • The velocity and trajectory of a satellite are crucial in maintaining its orbit.
  • Satellites can have circular or elliptical orbits, depending on their speed and altitude.
In the case of elliptical orbits, which are more complex than circular ones, satellites travel faster when closer to Earth (perigee) and slower when further away (apogee). This is due to the conservation of energy鈥攌inetic and potential energy exchange as the satellite moves through different points in its orbit.
Elliptical Orbits
Elliptical orbits are a specific type of orbit that follows an oval path, characterized by two main points: the perigee and the apogee. The perigee is the closest point to Earth, where the satellite travels fastest, and the apogee is the furthest, where it travels slowest.
  • Kepler's first law states that planets orbit the sun in elliptical paths, and the same applies to satellites around Earth.
  • The shape of the orbit is determined by its eccentricity. A higher eccentricity means a more elongated ellipse.
  • Elliptical orbits allow satellites to cover different altitudes during their path, which can be useful for satellite missions requiring varying distances.
Understanding elliptical orbits is crucial for calculating velocity changes and the timing of satellite operations, especially in tasks like GPS, communication, and weather monitoring.
Velocity at Apogee
The velocity at apogee is a key concept in understanding satellite motion in elliptical orbits. At the apogee, the satellite is at its highest point, and its velocity is at its minimum. This happens because the gravitational pull of the Earth has done maximum work, converting much of the kinetic energy back into potential energy.
Here鈥檚 how to compute it:
  • Using the conservation of energy, the total mechanical energy at apogee is equal to that at perigee.
  • The kinetic energy at apogee is lower because potential energy is at its highest.
  • By equating the mechanical energy expressions at perigee and apogee, we derive the velocity equation for apogee.
  • Given the mass and gravitational influences cancel in the energy balance equations, only the velocity and distance matter to find the velocity at apogee.
The velocity is crucial for understanding the satellite's stability in orbit and helps in planning maneuvers and adjustments to stay in the intended path.
Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy stored in an object due to its position in a gravitational field. For a satellite in orbit, this potential energy changes as it moves between perigee and apogee.
  • The formula for GPE is given by \(-\frac{GMm}{r}\), where \(G\) is the gravitational constant, \(M\) is Earth鈥檚 mass, \(m\) is the satellite鈥檚 mass, and \(r\) is the distance from the Earth鈥檚 center.
  • At perigee, due to minimal \(r\), the potential energy is lower compared to apogee where \(r\) is maximum and the potential energy is higher.
  • This interchange plays a crucial role in maintaining the satellite's motion and energy conservation in its orbit.
By understanding gravitational potential energy, we can better predict and control satellite movements, ensuring they function efficiently and predictably in their specific missions.

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Most popular questions from this chapter

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