HSC Physics Core 9.2 Space! Part 2: Launching into orbit! Overview of Part 2:!

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Steven Hutchinson
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1 Go to the ideo lesson for this slide deck: h2p://edrolo.com.au/subjects/physics/hsc- physics/space- part- 2/escape- elocity/lesson/ HSC Physics Core 9.2 Space Part 2: Launching into orbit Oeriew of Part 2: 2.1 Projectile motion (PM) 2.2 Escape elocity getting into orbit 2.3 Accelerations and g-forces during launch 2.4 Uniform circular motion (UCM) 2.5 Types of orbits 2.6 Kepler s Law of periods 2.7 Total orbital energy of a satellite 2.8 The sling-shot effect (graity-assist) 2.9 Orbital decay 2.10 Re-entry

2 HSC Physics Core 9.2 Space Part 2: Launching into orbit 2.2: Escape elocity Oeriew of Section 2.2: Escape elocity formula Velocity assist from Earth s axial rotation Velocity assist from Earth s orbital motion

3 Escape elocity Syllabus Module 9.2 dot point 2B4 Outline Newton s concept of escape elocity Syllabus Module 9.2 dot point 2B6 Discuss the effect of the Earth s orbital motion and its rotational motion on the launch of a rocket

4 Escape elocity Formula Escape elocity, u es the initial elocity (u) at the surface of the Earth that any object must be gien for it to just escape the Earth s graitational attraction. escape Earth s graitational attraction means the object will not fall back to the Earth or orbit the Earth. just escape means that just as the object reaches an infinite distance from the Earth its final elocity 0 By conseration of energy ( at surface) + ( at surface) ( at infinte distance) E E E k p p 1 2 m G m + rocket m Earth rocket u escape 2 0 r E So 2Gm Earth u escape r Exercise 1 Eeart

5 Escape elocity Exercise 1 Calculate the escape elocity from Earth. Data: Mass of Earth Radius of Earth 5.98 x kg 6380 km Uniersal Graitational Constant G 6.67 x N kg -2 m Answer

6 Escape elocity Exercise 1: Answer Calculate the escape elocity from Earth. Data: Mass of Earth 5.98 x kg Radius of Earth 6380 km Uniersal Graitational Constant G 6.67 x N kg -2 m 2 To find u es from the Earth 2GmEarth use ues R E (6.67x10 )(5.98x10 ) 6 (6.380x10 ) km s ms

7 Escape elocity Exercise 2 When a spacecraft is gien its escape elocity, (a) Does the direction in which it is launched matter (b) Does the path it follows matter? Answer

8 Escape Velocity Exercise 2: Answer (a) When a spacecraft is gien its escape elocity, (a) does the direction in which it is launched matter? Direction DOES matter: To sae fuel it is best to launch the ehicle towards the east to make use of the Earth s axial rotation towards the East. Answer (b)

9 Escape Velocity Exercise 2: Answer (b) When a spacecraft is gien its escape elocity, (b) Does the path it follows matter? Escape elocity is calculated by equating the kinetic energy at launch on the surface (r r surface ) with the gain in graitational potential energy the body would hae at infinity (r ) ( E ) k surface ΔE p ( E ) ( E ) p p surface Gm 0 Gmplanetm r planet planet r m planet spacecraft spacecraft So the kinetic energy needed at launch to escape the planet does not depend on the path the spacecraft follows. It depends only on the radius of the planet it is escaping Next Launch elocity-assist from Earth s axial-rotation

10 Launch elocity-assist from Earth s axial-rotation The size of the tangen&al speed of the Earth from east to west is different at different lactudes: θ equator cosθ This elocity can be used to assist a rocket at launch but the ehicle must be launched towards the East. Pr oof but so ω r ω constant ω θ Equator θ Equator so re requator rθ re θ θ ω Equator Equator r r θ E cosθ θ Example 1

11 Launch elocity-assist from Earth s axial-rotation Example 1 What is the tangential elocity of the Earth s surface at the Equator. Earth Data: Period of rotation 23 hours 56 minutes 4 seconds Radius of Earth 6371 km. Hint: If the Earth rotates uniformly on its axis, then: Equator 2π rearth T Earth Answer

12 Launch elocity-assist from Earth s axial-rotation Example 1: Answer Find tangential elocity of the Earth s surface at the Equator Earth data Period of rotation of Earth, T To find Use Equator Equator 2π r Earth Earth 3 2π m T Earth 3 2π m s m s Equator T Rotation E 23 h 56 min Radius of Earth, r 6371 km Conert units of T to seconds T Rotation Rotation 23 h 56 min s s s Exercise 3

13 Launch elocity-assist from Earth s axial-rotation Exercise 3: What is the tangential elocity of the Earth s surface at Sydney (latitude 34 0 S). Data -1 equator m s (from Example 1) θ sydney o 34 South Answer To find 34 o Sydney Sydney Use cosθ θ equator cos m s -1 o Exercise 4

14 Launch elocity-assist from Earth s axial rotation Exercise 4 Australia had a rocket launch site at Woomera in South Australia (latitude 30 o S) from the 1950s to 1970s. There hae been plans to build a new launch site on Cape York (latitude 12 0 S). Suppose you want to launch a rocket so that it will escape the Earth. Which location (Woomera or Cape York) would be the best site to locate the launch site? Explain why. Answer

15 Launch elocity-assist from Earth s axial rotation Exercise 4: Answer Which location (Woomera or Cape York) would be the best site to locate the launch site? Why? The tangential elocity of the Earth s surface at these two latitudes is calculated below. Earth's tangential elocity at Woomera Use latitude, θ Latitude θ 30 Woomera equator Woomera equator o cosθ m s x cos m s 0 Earth's tangential elocity at Cape York latitude, θ Cape York Cape York The best location will be the one with the greatest assistance from the Earth s rotational motion at launch. This occurs for the site with the smallest latitude (i.e. the site nearer the equator) Cape York. Next Launch elocity-assist from Earth s orbital motion Use Latitude θ 12 equator equator o cosθ m s x cos m s 0

16 Launch elocity-assist from Earth s orbital-motion about the Sun Once a ehicle is in orbit, the Earth s orbital elocity about the Sun is added to the ehicle s orbital elocity about the Earth. The Earth s orbital elocity about the sun V ES is about 30 km s -1 in an anti-clockwise direction r r r + rocket rocket's orbital elocity Earth 's orbital elocity relatie to Sun about the Earth about the Sun r r r + RS RE ES Exercise 5

17 Launch elocity-assist from Earth s orbital-motion about the Sun Exercise 5 Explain how you could take adantage of the Earth s orbital motion about the Sun to reach: (a) an outer planet one further from the Sun than the Earth (b) an inner planet one closer to the Sun than the Earth. Answer

18 Launch elocity-assist from Answer (a) Earth s orbital-motion about the Sun Exercise 5 Explain how you could take adantage of the Earth s orbital motion about the Sun to reach: (a) An outer planet one further from the Sun than the Earth Spacecraft heading to outer planets are not launched until the direction of the Earth in its orbit around the Sun corresponds with the desired direction. The spacecraft can then be launched up to a low Earth orbit, before firing its rockets again to accelerate ahead of the Earth The elocity boost receied from the Earth s orbital motion in this case is approximately 30 km s -1. Answer (b)

19 Launch elocity-assist from Answer (b) Earth s orbital-motion about the Sun Exercise 5 Explain how you could take adantage of the Earth s orbital motion about the Sun to reach: (b) an inner planet one closer to the Sun than the Earth. A spacecraft traeling from Earth to an inner planet will be naturally accelerated towards the planet by the Sun s graity. To assist the spacecraft, it should be: launched in the direction opposite to the orbital motion of the Earth about the Sun (i.e. in a clockwise direction): this will reduce its total orbital energy causing it to descent to a lower orbital radius about the Sun 1 GmM ET ( if ET becomes more negatie, ro ) decelerated (using its rocket engines) until it achiees a sun-orbit with a perihelion equal to the orbit of the inner planet. So, the spacecraft will continue to moe in the same direction as Earth, only more quickly. next ideo: Accelerations and g-forces 2 r o

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