xample: A 1-kg stone is thown vetically up fom the suface of the Moon by Supeman. The maximum height fom the suface eached by the stone is the same as the adius of the moon. Assuming no ai esistance and no gavitational foce except the one fom the Moon, what would be the initial speed? Radius of the Moon is 1700 km and the mass is 7.3x 10^ kg Stone Moon xample: A 1-kg stone is thown vetically up fom the suface of the Moon by Supeman. The stone neve comes back. Assuming no ai esistance and no gavitational foce except the one fom the Moon, what would be the condition fo the initial speed? Radius of the Moon is 1700 km and the mass is 7.3x 10^ kg Stone Moon 1
Consevation of mechanical enegy with gavitation mech detemines whethe motion is bound, fee, o at escape theshold mech is constant mech = K + Ug() Gmem U g() = always negative Fo mech < 0, paticle is bound and cannot escape. It cannot move beyond a tuning point (e.g., ) Fo mech > 0, paticle is fee. It can each = infinity and still have some K left mech = 0 is the escape condition. i A paticle at any location would need at least K = -U g () to move off to the ight and neve etun. U g =0 mech U( 1 ) FR K 1 BOUND 1 U g Gm = e Tuning point K = 0 Physics 106 Lectue 10 Keple s Laws and Planetay Motion SJ 7 th ed.: Chap 13.3, 13.6 Keple s laws of planetay motion Obit Law Aea Law Peiod Law Satellite and planetay obits Obits, potential, kinetic, total enegy
Goal Kelpe s 1 st and nd laws Johannes Keple (1571-1630, befoe Newton) Keple, a Geman mathematician and astonome, woked with Tycho Bahe, a geat obseve of planetay motion and inheited his data. Fom Bahe s data, Keple deduced 3 laws of planetay motion. Late, Newton deived and genealized Keples 3 laws fom the Newton s 3 laws and the law of univesal gavitational. Keple 3
Keple s 3 laws of planetay motion - summay 1. Obit Law: Planets all follow elliptical obits with the Sun at one focus Bound states, mech < 0 Cicle is special case Semi-majo majo axis =a a S. Aea Law: Angula momentum consevation The line fom the sun to a planet sweeps out equal aeas in equal time intevals Aeal velocity is constant da dt ΔA Δt aea swept time Δt ΔA S 1 ΔA Δt 3. Peiod Law: The squae of the obital peiod of any planet is popotional to the cube of the semi- majo axis of the elliptical obit 4π T = GM 3 Keple found these 3 laws fo planets aound the Sun. Newton showed they ae tue fo ANY pai of masses. 4
1. Obit Law: Obits (bound o not ) ae conic sections Mechanical negy detemines the type of obit K U () mech mech mech < > = 0 0 0 Conic Sections: Bound state Unbound state scape theshold lliptical o cicula obit Hypebolic obit (collision) Paabolic obit mech + llipse Hypebola Paabola g Deived fom law of gavitation 1 = +a + banch 1 f 1 f 1 + = a a 1 f a 1 f 1 - = -a -banch f The sum of the distances fom two fixed focal points is constant The diffeence of the distances fom two fixed foci is constant The pependicula distance fom a fixed line ( diectix ) equals the distance fom a fixed point ( focus ) a conic section : a cuve obtained by intesecting a cone with a plane 5
llipses The sum of 1 + is constant. F 1 and F ae each a focus of the ellipse, located a distance c fom the cente a = semi-majo axis, b = semi-mino axis The majo axis = a = 1 + aphelion The eccenticity is defined as e = c / a (c = ea) e is always < 1, e = 0 cicula obit, foci at cente Aea of ellipse = πab sun peihelion Planetay obits ae ellipses The sun is at one focus, the othe focus is empty Aphelion is the point futhest fom the sun Peihelion is the point closest to the sun Most planetay obits have low eccenticity Comets obits ae usually highly eccentic ath: e = 0.017 (vey small) peihelion = 91 x 10 6 miles aphelion = 95 x 10 6 miles Mecuy: e = 0.1 (most eccentic) Halley s Comet: e = 0.97. Aea Law: quivalent to angula momentum consevation L is constant since gavity cannot exet a toque aound the cental mass M da aea swept out duing time dt 1 1 = base height = v dt 1 1 = mv dt = L dt m m So, p = mv ds v dt slow aphelion v m x v φ Tiangle aea da M fast peihelion " aeal velocity" da = dt = ate at which adius sweeps out aea L m = constant at all points of the obit Compae speeds at aphelion (fa) and peihelion (close): Lpeihelion = mp vp, = mp vp v a p = < 1, so v aphelion < v L = m v = m v v aphelion a a, a a p a peihelion 6
Apogee and peigee 10.1. An ath satellite is placed in an obit whose apogee distance is 9 times its peigee distance. The atio of the speed at peigee to the speed at apogee is: A) 10 B) 9 C) 1/9 D) 1/10 ) 3 slow apogee x apo pei M fast peigee L = mv = L = mv a a a p p p 7