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6.6 Motion of Orbiting Satellites

Syllabus
2024
Topic
6.6
Level
—

Learning objectives

6.6A—Describe the motions of a system consisting of two objects interacting only via gravitational forces. AP Physics…Describe the motions of a system consisting of two objects interacting only via gravitational forces. AP Physics 1: Algebra-Based Course and Exam Description THIS PAGE IS INTENTIONALLY LEFT BLANK. AP PHYSICS 1 UNIT Oscillations 5–8% AP EXAM WEIGHTING ~5–10 CLASS PERIODS 7 119 119 | AP Physics 1: Algebra-Based Course and Exam Description Remember to go to AP Classroom to assign students the online Progress Check for this unit. Whether assigned as homework or completed in class, the Progress Check provides each student with immediate feedback related to this unit’s topics and science practices. Progress Check 7 Multiple-choice: ~18 questions Free-response: 4 questions• In a system consisting only of a massive central object and an orbiting satellite with mass that is negligible in comparison to the central object’s mass, the motion of the central object itself is negligible.• The motion of satellites in orbits is constrained by conservation laws.- i. In circular orbits, the system’s total mechanical energy, the system’s gravitational potential energy, and the satellite’s angular momentum and kinetic energy are constant.- ii. In elliptical orbits, the system’s total mechanical energy and the satellite’s angular momentum are constant, but the system’s gravitational potential energy and the satellite’s kinetic energy can each change.- iii. The gravitational potential energy of a system consisting of a satellite and a massive central object is defined to be zero when the satellite is an infinite distance from the central object. Relevant equation: mmUGg =− 12 r• The escape velocity of a satellite is the satellite’s velocity such that the mechanical energy of the satellite–central-object system is equal to zero.- i. When the only force exerted on a satellite is gravity from a central object, a satellite that reaches escape velocity will move away from the central body until its speed reaches zero at an infinite distance from the central body.- ii. The escape velocity of a satellite from a central body of mass M can be derived using conservation of energy laws. Derived equation: 2GMvesc = r

Use conservation laws to explain satellite motion

Choose the two-body model

Model a satellite of mass mm interacting only with a much more massive central body of mass MM. The central body's motion is then negligible, while gravity transfers energy between kinetic and gravitational potential forms without changing the system's total mechanical energy.

Compare circular and elliptical orbits

Quantity Circular orbit Elliptical orbit
System total mechanical energy Constant Constant
Satellite angular momentum Constant Constant
Gravitational potential energy Constant Changes with radius
Satellite kinetic energy and speed Constant Change around the orbit

Use the gravitational energy reference

Ug=−GMmr(Ug=0 at r→∞)\begin{gathered}U_g=-\frac{GMm}{r}\\(U_g=0\text{ at }r\to\infty)\end{gathered}

Trace energy around an ellipse

In an elliptical orbit, moving closer makes UgU_g more negative. Because E=K+UgE=K+U_g stays constant, KK increases and the satellite moves faster. Moving farther away reverses the exchange. Angular momentum remains constant throughout.

Derive the minimum escape speed

Minimum escape condition: choose the satellite–central-body system. At launch radius rr, set its total mechanical energy to zero so that at infinite distance both UgU_g and the final speed approach zero.

0=12mvesc2−GMmr0=\tfrac12mv_{\mathrm{esc}}^2-\dfrac{GMm}{r}

12mvesc2=GMmr\tfrac12mv_{\mathrm{esc}}^2=\dfrac{GMm}{r}

vesc=2GMrv_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{r}}.

The satellite mass cancels; for the same central body, doubling rr reduces escape speed by a factor of 2\sqrt2.

Interpret escape correctly

Escape velocity is the initial speed for zero total mechanical energy under gravity alone. It is not a speed maintained during escape: at the minimum value, the satellite slows toward zero speed only as rr approaches infinity. Atmosphere, thrust, and other forces are outside this model.