Perpetual Freefall (Orbit) - Revision history

Laurence12799: /* Examples */

2019-07-04T01:30:53Z

Examples

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	===Simple===		===Simple===
	~~'''INSERT DRAWING'''~~		[[File:PFSimpleSketch.jpeg\|300px\|right\|thumb\|A diagram of the simple example]]

	A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> from the surface of the Earth and speed <math>v_T</math>.		A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> from the surface of the Earth and speed <math>v_T</math>.
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	===Middling===		===Middling===
			[[File:PFMiddlingSketch.jpeg\|200px\|right\|thumb\|A diagram of the middling example]]

	A rocket of mass <math>m</math> is on the surface of the moon. The mass of the moon is <math>M_m = 7.36 \times 10^{22} \ \text{kg}</math>. The radius of the moon is <math>R_m = 1.74 \times 10^6 \ \text{meters}</math>.		A rocket of mass <math>m</math> is on the surface of the moon. The mass of the moon is <math>M_m = 7.36 \times 10^{22} \ \text{kg}</math>. The radius of the moon is <math>R_m = 1.74 \times 10^6 \ \text{meters}</math>.

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	===Difficult===		===Difficult===
	~~'''INSERT SKETCH'''~~		[[File:PFDifficultSketch.jpeg\|400px\|right\|thumb\|A diagram of the difficult example]]

	A satellite of mass <math>m_1</math> is in an elliptical orbit around a planet of mass <math>m_2</math>, whose center is at F, one of the two foci of the ellipse (<math>m_2 >> m_1</math>). P is the closest approach of the two masses and is a distance <math>r_p</math> from the foci F. The satellite's speed at the point P is <math>v_p</math>. The two ,asses are furthest apart at point A, which is a distance <math>r_A = 5r_p</math>.		A satellite of mass <math>m_1</math> is in an elliptical orbit around a planet of mass <math>m_2</math>, whose center is at F, one of the two foci of the ellipse (<math>m_2 >> m_1</math>). P is the closest approach of the two masses and is a distance <math>r_p</math> from the foci F. The satellite's speed at the point P is <math>v_p</math>. The two ,asses are furthest apart at point A, which is a distance <math>r_A = 5r_p</math>.

Laurence12799: /* See also */

2019-07-04T00:50:26Z

Laurence12799 at 00:49, 4 July 2019

2019-07-04T00:49:42Z

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	:[http://www.braeunig.us/space/orbmech.htm Orbital Mechanics]		:[http://www.braeunig.us/space/orbmech.htm Orbital Mechanics]

			===External Links===

			==References==

	[[Category:Interactions]]		[[Category:Interactions]]

Laurence12799: /* See also */

2019-07-04T00:49:01Z

Laurence12799: /* Simple */

2019-07-03T23:35:18Z

Simple

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	'''INSERT DRAWING'''		'''INSERT DRAWING'''

	A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> and speed <math>v_T</math>.		A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> from the surface of the Earth and speed <math>v_T</math>.

	:'''a) Find an expression for the tangential velocity (<math>v_T</math>) of the satellite?'''		:'''a) Find an expression for the tangential velocity (<math>v_T</math>) of the satellite?'''

Laurence12799: /* Simple */

2019-07-03T23:35:02Z

Simple

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	'''INSERT DRAWING'''		'''INSERT DRAWING'''

	A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> and speed v.		A satellite of mass <math>m</math> has a circular orbit about the Earth at a radius <math>r</math> and speed <math>v_T</math>.

	:'''a) Find an expression for the tangential velocity (<math>v_T</math>) of the satellite?'''		:'''a) Find an expression for the tangential velocity (<math>v_T</math>) of the satellite?'''

Laurence12799: /* Difficult */

2019-07-03T19:31:31Z

Difficult

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	:::<math>v_A = \frac{r_p}{5}</math>		:::<math>v_A = \frac{r_p}{5}</math>


	:'''b) What is the energy of the satellite '''(<math>E_{satellite}</math>)'''?'''		:'''b) What is the energy of the satellite '''(<math>E_{satellite}</math>)'''?'''

Laurence12799: /* Difficult */

2019-07-03T19:31:19Z

Difficult

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	::Note the energy of the satellite is less than 0 as it should be for a satellite in orbit.		::Note the energy of the satellite is less than 0 as it should be for a satellite in orbit.

	~~[[File:FinalEX.png]]~~

	~~[[File:FinalEX2.png]]~~

	~~[[File:FinalEX3.png]]~~

	In compliance with MIT OpenCourseWare privacy and terms, the previous example maybe found here [https://ocw.mit.edu/high-school/physics/exam-prep/oscillations-gravitation/orbits-of-planets-satellites/8_01x_fall_2002_final_pract_sol.pdf]		In compliance with MIT OpenCourseWare privacy and terms, the previous example maybe found here [https://ocw.mit.edu/high-school/physics/exam-prep/oscillations-gravitation/orbits-of-planets-satellites/8_01x_fall_2002_final_pract_sol.pdf]

Laurence12799: /* Difficult */

2019-07-03T19:30:51Z

Difficult

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	:'''b) What is the energy of the satellite '''(<math>E_{satellite}</math>)'''?'''		:'''b) What is the energy of the satellite '''(<math>E_{satellite}</math>)'''?'''

			::Since we aren't taking heat into consideration, the energy we wish to find is the kinetic energy plus the potential energy:

			:::<math>E = K + U</math>

			::The only force working on the satellite is gravity, so the only potential energy is due to gravity, which we found a potential energy function for in the middling example:

			:::<math>E = K - \frac{Gm_1m_2}{r} = \frac{1}{2}mv^2 - \frac{Gm_1m_2}{r}</math>

			::We know energy should be conserved in the elliptical orbit, so the energy at point P should equal the energy at point A:

			:::<math>E_p = E_A</math>

			::Which is:

			:::<math>\frac{1}{2}m_1 {v_p}^2 - \frac{Gm_1m_2}{r_p} = \frac{1}{2}m_1 {v_A}^2 - \frac{Gm_1m_2}{r_A}</math>

			::Plugging in <math>5v_A \ \text{for} \ v_p \ \text{and} \ 5r_p \ \text{for} \ r_A</math> gives:

			:::<math>\frac{1}{2}m_1 {(5v_A)}^2 - \frac{Gm_1m_2}{r_p} = \frac{1}{2}m_1 {v_A}^2 - \frac{Gm_1m_2}{5r_p}</math>

			::Moving terms around creates:

			:::<math>\frac{1}{2}m_1 {(5v_A)}^2 - \frac{1}{2}m_1 {v_A}^2 = \frac{Gm_1m_2}{r_p} - \frac{Gm_1m_2}{5r_p}</math>

			::And combining like terms and factoring gives:

			:::<math>12m_1 v_A^2 = \frac{4Gm_1m_2}{5r_p}</math>

			::Cancelling <math>m_1</math> on each side dividing each side by 4 gives:

			:::<math>3{v_A}^2 = \frac{Gm_2}{5r_p}</math>

			::Thus:

			:::<math>v_A = \sqrt{\frac{Gm_2}{15r_p}}</math>

			::We can plug this value of <math>v_A</math> back into the energy equation for point P, since <math>5v_A = v_p</math> to get:

			:::<math>E_p = \frac{1}{2}m_1 \left(5\sqrt{\frac{Gm_2}{15r_p}}\right)^2 - \frac{Gm_1m_2}{r_p}</math>

			::Which equals:

			:::<math>E_p = \frac{25}{2} \frac{Gm_1m_2}{15r_p} - \frac{Gm_1m_2}{r_p}</math>

			:::<math>E_p = \frac{Gm_1m_2}{r_p} \left(\frac{5}{6} - \frac{6}{6}\right)</math>

			::Therefore the energy of the satellite is:

			:::<math>E = -\frac{1}{6}\frac{Gm_1m_2}{r_p} \lt 0</math>

			::Note the energy of the satellite is less than 0 as it should be for a satellite in orbit.

	[[File:FinalEX.png]]		[[File:FinalEX.png]]

Laurence12799: /* Difficult */

2019-07-03T18:56:26Z

Difficult

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 ===Difficult===
 A satellite of mass <math>m_1</math> is in an elliptical orbit around a planet of mass <math>m_2</math>, whose center is at F, one of the two foci of the ellipse (<math>m_2 >> m_1</math>). P is the closest approach of the two masses and is a distance <math>r_p</math> from the foci F. The satellite's speed at the point P is <math>v_p</math>. The two ,asses are furthest apart at point A, which is a distance <math>r_A = 5r_p</math>.
 :'''a) What is the speed of the satellite at point A '''(<math>v_A</math>)'''?'''
 :'''b) What is the energy of the satellite '''(<math>E_{satellite}</math>)'''?'''

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	:[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]		:[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]
	:[http://www.physicsbook.gatech.edu/Projectile_Motion Projectile Motion]<br>		:[http://www.physicsbook.gatech.edu/Projectile_Motion Projectile Motion]<br>
	:		:[[Curving Motion]]

	===Further reading===		===Further reading===

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	== See also ==		== See also ==
	[[http://www.physicsbook.gatech.edu/Law_of_Gravitation \| Law of Gravitation]]		:[http://www.physicsbook.gatech.edu/Law_of_Gravitation Law of Gravitation]
	[[http://www.physicsbook.gatech.edu/Newton%27s_Laws \| Newton's Laws]]		:[http://www.physicsbook.gatech.edu/Newton%27s_Laws Newton's Laws]
	[[http://www.physicsbook.gatech.edu/Momentum_Principle \| Momentum Principle]]		:[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]
	[[http://www.physicsbook.gatech.edu/Projectile_Motion \| Projectile Motion]]		:[http://www.physicsbook.gatech.edu/Projectile_Motion Projectile Motion]<br>
			:

	===Further reading===		===Further reading===

	[[https://books.google.com/books?id=1kUkBAAAQBAJ&pg=PA106&lpg=PA106&dq=perpetual+freefall&source=bl&ots=7RAUQiUgxl&sig=e6gf1UcNWgCGVIoEgq-iTlpuUuY&hl=en&sa=X&ved=0ahUKEwiq3JritcbJAhWKLB4KHeuRCM8Q6AEIdTAR#v=onepage&q=perpetual%20freefall&f=false \| The Observer's Guide to Planetary Motion]]		:[https://books.google.com/books?id=1kUkBAAAQBAJ&pg=PA106&lpg=PA106&dq=perpetual+freefall&source=bl&ots=7RAUQiUgxl&sig=e6gf1UcNWgCGVIoEgq-iTlpuUuY&hl=en&sa=X&ved=0ahUKEwiq3JritcbJAhWKLB4KHeuRCM8Q6AEIdTAR#v=onepage&q=perpetual%20freefall&f=false The Observer's Guide to Planetary Motion]

	[https://phet.colorado.edu/en/simulation/legacy/gravity-and-orbits [Orbit Simulation]]		:[https://phet.colorado.edu/en/simulation/legacy/gravity-and-orbits Orbit Simulation]

	[http://www.braeunig.us/space/orbmech.htm [Orbital Mechanics]]		:[http://www.braeunig.us/space/orbmech.htm Orbital Mechanics]

	[[Category:Interactions]]		[[Category:Interactions]]