AE 245 homework #9 solutions

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1 AE 245 homewok #9 olution Tim Smith 13 Apil Poblem1 In the Apollo miion fom the Eath to the Moon, the Satun thid tage povided the tan-luna inetion bun that tanfeed the Apollo pacecaft fom a low Eath obit at an altitude of 185 into a paabolic obit that eached a luna altitude of 300. Thi wa followed by a luna obit inetion bun that placed the Apollo pacecaft into an elliptical obit about the Moon with maximum luna altitude of 300 and minimal luna altitude of 100. Fo thi pat of the Apollo miion, invetigate a Hohmann tanfe followed by a luna obit inetion bun that would have tanfeed the Apollo pacecaft fom it initial low Eath obit into the ame elliptical obit about the Moon. 1. Decibe a Hohmann tanfe that would tanfe the Apollo pacecaft fom low Eath cicula obit at an altitude of 185 into a cicula Eath obit with luna adiu. What ae the 4V fo the two impulive bun? Decibe the elliptical tanfe obit. 2. Auming the cicula Eath obit with luna adiu i achieved a in pat (a) o that the Apollo pacecaft eache a luna altitude of 300, what i the 4V that would inet the Apollo pacecaft into an elliptical obit about the Moon with maximum luna altitude of 300 and minimal luna altitude of 100? 3. Fo thi pat of the Apollo miion, dicu the advantage and diadvantage of uing the Hohmann tanfe appoach compaed with the paabolic tanfe appoach that wa actually ued. 1.1 Eath-to-Moon Hohmann tanfe The Hohmann ellipe ha peigee p0 (6378: ) and apogee a0 em 384;400. The peigee bun ha poigade delta-vee 4v 1 2µ e p0 + a0 a0 p0, p0 2[398;600] 6563: ; ;400, 6563:1! 398;600 3: :1 while the apogee bun ha poigade delta-vee 4v 2 a0, 2µ e p0 p0 + a0 a0 2[398;600] 6563: ; :1, 384; ; ;400! 0:

2 1.2 Luna inetion Having eached a cicula Eath obit at the Moon obital adiu, the pacecaft i at et with epect to the Moon. Thu, the delta-vee fo inetion at apocynthion 1 p1 (1737: ) into an elliptical obit with peicynthion a0 (1737: ) i 4v 3 v 3 2µ m p1 p1 + a1 a1 2[4902:8] 2037: :4 1837:4 1: :4 1.3 Hohmann v paabolic tanfe Since I didn t take vey good note duing lectue, I ll ecapitulate the paabolic tanfe method. A paabolic obit ha zeo total enegy, E 0, o the pacecaft peed elative to the Eath at luna obit v ce 2µ e em 2[398;600] 384;400 1:4401 The tangential component of thi velocity v p 0 p0 p 0 2µ e 6563:1 2[398;600] ( θ) ce a0 a0 p0 384; :1 0:18817 o the adial component ṙ ce v 2 ce, ( θ) 2 ce [1:4401] 2, [0:18817] 2 1:4278 The Moon ha a tangential velocity of magnitude 398;600 v me em 384;400 1:0182 o the pacecaft velocity elative to the Moon (in Eath-centeed pola coodinate) ~v cm ~v ce,~v me (1:4278ê + 0:18817ê θ, 1:0183ê θ ) (1:4278ê, 0:83013ê θ ) ha the magnitude v cm j~v cm j [1:4278] 2 +[0:83013] 2 1:6515 Thu, the delta-vee euied to go fom paabolic Eath obit to elliptical luna obit i etogade, 1 Fathet point in the obit fom the Moon. 2

3 4v 4 v 3, v cm (1:5105, 1:6516),140:10 m The total delta-vee euied fo the paabolic tanfe p 398;600 4v p j4v 4 j + j( 2, 1) j 0: :41421 p0 6563:1! 3:3691 while the total delta-vee euied fo the Hohmann + inetion tanfe 4v h j4v 1 j + j4v 2 j + j4v 3 j (3: : :5105) 5:6171 i about 67% lage. 2 Poblem2 A pacecaft i in a 300 altitude cicula obit about Ma. 1. What i the velocity of the pacecaft in thi cicula obit? 2. What i the minimal 4V euied fo the pacecaft to ecape the gavitation of Ma? 3. What i the minimal 4V euied fo the pacecaft to ecape the gavitation of Ma and of the Sun? 2.1 Ma cicula obit In a cicula obit at 0 (3389: ) , the cicula peed µm 42;850 v c :3 3: Ma paabolic obit Minimal ecape fom Ma gavitational influence occu on a paabolic tajectoy, with delta-vee 2µm µm 4v 1, (p 2, 1)vc 0:41421(3:4080) 0 0 1:4116 3

4 2.3 Ma ecape to ola paabolic obit Ecape fom both Ma and the Sun gavity euie hypebolic ecape, with 2µm µm 4v 2 +(v cm ) 2, 2v 0 2 c +(v cm )2, v c 0 whee (v cm ) i the hypebolic exce peed with epect to Ma. Fo ecape fom the ola ytem, we ue a poigade exit, o that the pacecaft peed with epect to the Sun (v c ) (v cm ) + v m whee the peed of Ma aound the Sun µ 1: v 11 m m 2: :137 Minimal ecape fom the ola ytem euie a paabolic obit aound the Sun, (v c ) 2µ m p 2 vm 34:134 o the hypebolic exce velocity (v cm ) (v c ), v m (31:134, 24:137) 9:997 Thu, the euied delta-vee fo ecape fom Ma and the Sun gavity 4v 2 2[3:4080] 2 +[9:997] 2, 3:4080 7: Poblem3 An inteplanetay pacecaft miion fom Eath to Jupite i popoed. 1. Find the 4V euied at a 1.1 Eath adii cicula paking obit about the Eath that will place the pacecaft on a Hohmann tanfe elliptical obit that connect with the cicula heliocentic obit of Jupite. Include in you analyi the effect of the Eath gavitation. What i the tanfe time fom Eath to Jupite? 2. Find the 4V euied at a 1.1 Eath adii cicula paking obit about the Eath that will place the pacecaft on a paabolic heliocentic obit that inteect the cicula heliocentic obit of Jupite. Include in you analyi the effect of the Eath gavitation. 4

5 3.1 Eath ecape to Eath-Jupite Hohmann tanfe Stating at 0 1:1(6378:1) , the initial cicula peed v ; :9 7:5375 while the hypebolic ecape velocity with epect to the Eath v 1 2v 2 0 +(v ce )2 Since the Eath obit the Sun at µ 1: v 11 e e 1: :789 and the peihelion peed fo an ellipe with p e 1: and a j 7: i v p (v c ) 2µ a p + a p 2[1: ] 7:7819 [1: :7819]10 8 1: :582 the hypebolic exce peed (fo a poigade exit) i (v ce ) (v c ), v e (38:582, 29:789) 8:7930 Thu, the euied delta-vee 4v A 2v 2 0 +(v ce) 2, v 0 2[7:5375] 2 +[8:7930] 2, 7:5375 6:2805 The tanfe time i, a fo all Hohmann tanfe, the half-peiod t π ( p + a ) 3 ([1: :7819]10 π 8 ) 3 8µ 8(1: : y 266 day 13:6h ) 3.2 Eath ecape to ola paabolic obit Fo paabolic heliocentic obit, the peihelion peed with epect to the Sun (v c ) 2µ e p 2 ve 42:128 5

6 o the hypebolic exce peed (v ce ) (v c ), v e (42:128, 29:789) 12:339 which euie a delta-vee 4v B 2v 2 0 +(v ce) 2, v 0 2[7:5375] 2 +[12:339] 2, 7:5375 8: Poblem4 An ateoid that ha one-tenth the ma of the Moon tike the Moon. The ateoid i moving towad the Moon along the line joining the cente of the two bodie with a velocity that i eual in magnitude to the velocity of the Moon but oppoite in diection. In the colliion, the entie momentum of the ateoid i tanfeed to the Moon. 1. What i the velocity of the Moon befoe and afte the impact of the ateoid on the Moon? 2. Afte the colliion, the Moon will be in an elliptic obit aound the Eath. Detemine the emi-majo axi of the new obit and alo the eccenticity of the new obit. 4.1 Inelatic colliion The linea momentum balance in the +ê θ diection i M m v 0, M a v 0 (M m + M a )v 1 whee the pe-colliion velocity (a in poblem 1(a)) i ~v 0 ê θ 1:0183ê θ em o the pot-colliion velocity ~v 1 M m, M a M m + M a ~v 0 1, M am m 1 + M a M m ~v 0 1, 0: :1 (1:0183ê θ) 0: Pot-colliion obit The new obit ha a pecific total enegy E v2 1, µ e [0:83315] 2, [1:0183] 2 2 em 2 2,0:

7 o the emi-majo axi a 1, µ e, 398; ;900 2E 1 2[,0:68986] A the apogee adiu emain a em a(1 + e) 384;400 the eccenticity of the new obit e a a, 1 384; ;900, 1 0: Poblem5 Conide a ingle tage launch vehicle with a chemical ocket having pecific impule of 311 ec and ocket engine exhaut velocity of 3048 m/ec. I the launch vehicle capable of launching a payload into low Eath obit. Aume the bun time of the ocket engine i 250 ec. and the ma of the launch vehicle i 90 pecent popellant. Since the launch vehicle i 90% popellant, it final ma i 10% of it initial ma, o it ma atio M 0 100% M 1 10% 10 Auming vetical launch, the maximum available delta-vee 4v v e ln M 0 M 1, g o τ b (3048ln10, 9:807[250]) m 4566:5 m which in t enough (aound 7.9 /) to get into LEO. 6 Poblem6 A chemical ocket having pecific impule of 311 ec and ocket engine exhaut velocity of 3048 m/ec can be ued to change the velocity vecto of a pacecaft in a cicula LEO at an altitude of What i the maximum 4V that can be achieved uing thi ocket engine? Aume the bun time of the ocket engine i 20 and the ma of the pacecaft in LEO i 50 pecent popellant. 2. Can the pacecaft ecape fom the phee of influence of the Eath uing thi ocket engine? If o, what i the pacecaft teminal velocity? If not, what i the maximum altitude cicula obit that the pacecaft can each? 7

8 6.1 Maximum 4V, tangential bun A befoe, 50% popellant ma mean the ma atio M 0 M 1 2. In a cicula obit, the mot efficient tanfe obit i a Hohmann tanfe; ince thi euie hoizontal bun at peigee and apogee, the pacecaft i not gaining potential enegy duing the bun, and the maximum delta-vee i 4v v e ln M 0 M 1 (3048ln2) m 2112:7 m 6.2 Boot to high obit Eath ecape euie a delta-vee 4v ec p2, 1 0 0: ; :1 3200:1 o the available delta-vee in t ufficient fo ecape. A Hohmann tanfe euie 4v v p, v 0 + v 1, v a whee peigee peed v p 2µ e and apogee peed v a 2µ e ae function of apogee adiu 1, but the initial cicula peed v ; :1 7:7258 i not. Speadheet iteation how that uing the available 4v in a two-bun Hohmann tanfe can lift the pacecaft to a cicula obit at 1 12;911 8

HW6 Physics 311 Mechanics

HW6 Physics 311 Mechanics Fall 015 Physics depatment Univesity of Wisconsin, Madison Instucto: Pofesso Stefan Westehoff By Nasse M. Abbasi June 1, 016 Contents 0.1 Poblem 1.........................................