#AP Physics C: Mechanics - Satellite Motion 🚀

Hey there! Let's get you prepped for the AP Physics C: Mechanics exam with a deep dive into satellite motion. We'll break down everything from circular orbits to escape velocity, making sure you're not just memorizing formulas but truly understanding the concepts. Let's get started!

#Motions in Gravitational Systems

#Central Object Motion

• Key Idea: In a satellite-central object system, the central object is so much more massive that we treat it as stationary. 🪐
• The central object's motion is negligible compared to the satellite's orbit.
• No need to calculate the central object's motion; it's effectively at rest.

#Satellite Orbit Constraints

• Conservation Laws: These are your best friends! They dictate how satellites move.
• Circular Orbits:
  • Total mechanical energy, gravitational potential energy, kinetic energy, and angular momentum are all constant.
  • Think of it as a perfectly balanced system.
• Elliptical Orbits:
  • Total mechanical energy and angular momentum are constant.
  • Gravitational potential energy and kinetic energy vary throughout the orbit.
• Gravitational Potential Energy:
  • Defined as zero when the satellite is infinitely far away.
  • Becomes increasingly negative as the satellite gets closer to the central object.

#Energy in Circular Orbits

• Energy Relationship: The total energy can be expressed using either potential or kinetic energy.
• Kinetic Energy: $K = -\frac{1}{2} U$ (always half the magnitude of potential energy).
• Total Energy: $E_{\text{total}} = \frac{1}{2} U = -\frac{GMm}{2r}$ 🛰️
  • $G$ = gravitational constant
  • $M$ = mass of the central object
  • $m$ = mass of the satellite
  • $r$ = radius of the circular orbit

#Energy in Elliptical Orbits

• Varying Distance: The satellite's distance from the central object changes throughout the orbit.
• Constant Values: Total mechanical energy and angular momentum are conserved.
• Periapsis:
  • Closest approach.
  • Highest kinetic energy.
  • Lowest (most negative) gravitational potential energy.
• Apoapsis:
  • Farthest point.
  • Lowest kinetic energy.
  • Highest (least negative) gravitational potential energy.

#Escape Velocity

• Definition: The minimum speed needed to break free from the central object's gravity. 🚀
• Total Mechanical Energy: Becomes zero at escape velocity.
• Path: If gravity is the only force, the satellite will continue moving away until its speed is zero at an infinite distance.
• Calculation:
  • $\frac{1}{2}mv_{\text{escape}}^2 - \frac{GMm}{r} = 0$
  • $m$ = satellite's mass
  • $v_{\text{escape}}$ = escape velocity
  • $G$ = gravitational constant
  • $M$ = central object's mass
  • $r$ = distance between satellite and central object

#Final Exam Focus 🎯

• High-Priority Topics:
  • Energy conservation in circular and elliptical orbits.
  • Relationship between kinetic and potential energy.
  • Escape velocity calculations.
• Common Question Types:
  • MCQs involving energy changes and orbital mechanics.
  • FRQs requiring detailed analysis of energy conservation and satellite motion.
  • Questions that combine concepts from multiple units, like circular motion and gravitation.
• Time Management:
  • Quickly identify the core concepts in each problem.
  • Use conservation laws to simplify calculations.
  • Don't get bogged down in complex algebra; focus on the physics.
• Common Pitfalls:
  • Incorrectly applying energy conservation principles.
  • Forgetting the negative sign for gravitational potential energy.
  • Mixing up circular and elliptical orbit characteristics.
• Strategies for Challenging Questions:
  • Draw diagrams to visualize the problem.
  • Break down the problem into smaller, manageable steps.
  • Check your units and make sure your answers make sense.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. A satellite is in a circular orbit around a planet. If the radius of the orbit is doubled, what happens to the satellite's orbital speed? (A) It increases by a factor of 2 (B) It decreases by a factor of 2 (C) It increases by a factor of $\sqrt{2}$ (D) It decreases by a factor of $\sqrt{2}$

2. A satellite moves in an elliptical orbit around a planet. At which point in the orbit is the satellite's kinetic energy the greatest? (A) At the point farthest from the planet (B) At the point closest to the planet (C) When the satellite is moving perpendicular to the radius (D) Kinetic energy is constant throughout the orbit

3. What is the minimum speed needed for a satellite to escape the gravitational pull of a planet? (A) Orbital speed (B) Escape velocity (C) Circular velocity (D) Zero velocity

#Free Response Question

A satellite of mass m is in an elliptical orbit around a planet of mass M. At its periapsis (closest approach), the satellite is at a distance r from the planet's center and has a speed of v. The gravitational constant is G.

(a) Derive an expression for the total mechanical energy of the satellite-planet system at the periapsis in terms of G, M, m, r, and v.

(b) Derive an expression for the angular momentum of the satellite about the center of the planet at the periapsis in terms of m, r, and v.

(c) At the apoapsis (farthest point), the satellite is at a distance of 2r from the planet's center. Determine the speed of the satellite at the apoapsis in terms of v.

(d) Determine the total mechanical energy of the system at the apoapsis in terms of G, M, m, r, and v.

#FRQ Scoring Guide

(a) Total Mechanical Energy at Periapsis (3 points)

• 1 point: Correct expression for kinetic energy: $K = \frac{1}{2}mv^2$
• 1 point: Correct expression for potential energy: $U = -\frac{GMm}{r}$
• 1 point: Correct expression for total mechanical energy: $E = \frac{1}{2}mv^2 - \frac{GMm}{r}$

(b) Angular Momentum at Periapsis (2 points)

• 1 point: Correct understanding of angular momentum: $L = r \times p$
• 1 point: Correct expression for angular momentum: $L = mvr$

(c) Speed at Apoapsis (4 points)

• 1 point: Realizing angular momentum is conserved: $L_{\text{periapsis}} = L_{\text{apoapsis}}$
• 1 point: Correct expression for angular momentum at apoapsis: $L_{\text{apoapsis}} = m(2r)v_{\text{apoapsis}}$
• 1 point: Equating angular momentum at periapsis and apoapsis: $mvr = m(2r)v_{\text{apoapsis}}$
• 1 point: Correct expression for speed at apoapsis: $v_{\text{apoapsis}} = \frac{v}{2}$

(d) Total Mechanical Energy at Apoapsis (2 points)

• 1 point: Realizing total mechanical energy is conserved: $E_{\text{periapsis}} = E_{\text{apoapsis}}$
• 1 point: Correctly stating the total mechanical energy at apoapsis is the same as at periapsis: $E_{\text{apoapsis}} = \frac{1}{2}mv^2 - \frac{GMm}{r}$

Good luck, you've got this! 💪