Gravitational Potential Energy and Space Travel
Explore the concepts of gravitational potential energy, orbital energy, escape speed, and satellite motion in the context of space travel. Learn about critical escape conditions, escape speeds from Earth and orbit, and the speed of satellites in circular orbits. Delve into multiple objects in space and their interactions to grasp the fundamentals of energy in the universe.
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Presentation Transcript
Lecture 14: Gravitational potential energy and space travel Universal gravitational potential energy Space travel problems Escape speed Orbital energy Multiple objects
Gravitational potential energy ?????=??? Conservative force ?2, attractive ?? ? ? ? ?? ??= ?? ?= ??
?? ? ? ? ?? ??= ?? ?= ?? ?? ? ? ? = ?? ?? ??? ?2 ?? ???? = ?? ?? ?? ?? = ??? =??? ??? ? ?? ?? ?? 1 ?? 1 ?? ??= ??? ?? ?????= ??? Choose reference point: ?0= . Assign ? ?0= = 0 ? negative!
Potential energy diagram ? ?0= = 0
Escape condition ? = ??2 ??? ? Critical escape condition: barely making it to ? = ????????, having slowed to speed of zero
Escape speed Minimum speed an object must have at distance R from central mass M if it is to go infinitely far away ??= ?? ??? 2 ??? ? = 2 ????? = ?? ? ??? 2 ????? = 0 0 ? 2?? ? ????=
Example: Escape speed from Earth ????? = 5.97 1024?? ????? = 6.38 106 ? ? = 6.67 10 11? ?2/??2 2?? ? ????= = 11,200 ?/?
Example: Escape speed from orbit Speed necessary to escape gravitational field of the Sun when object is launched from Earth: ????=1.99 10 30 kg ???= 1.50 10 11 m ?= 6.67 10 -11 N m2/kg2 2????? ??? ????= = 42,000 ?/?
Orbital Energy ? = ? + ? = ??2 ??? ? Speed of satellite in circular orbit: ?2=?? ? ?? ? ??? ? = ? ? ? = ??? 2?
Satellite Motion ??= ?????,?= ??????=????(+??) ?2 ? ?????? ?2 = ???? ?? ? = ?2
Example with multiple objects A planet has mass 4? and a radius 2?. Its moon has mass ? and radius ?. The centers of the planet and moon are a distance 9? apart. A shuttle of mass ? is a distance 4? away from the center of the planet and moving with speed ?. What is the total mechanical energy of the shuttle? V 4M M r m 2r 4r 9r
If the shuttle was initially at rest at X, how much work did the engines do? V 4M M r m 2r 4r 9r
