Slutins t Prblems. Nttin: V speed f cmet immeditely befre cllisin. V speed f cmbined bject immeditely fter cllisin, mmentum is cnserved. V, becuse liner + k q perihelin distnce f riginl prblic rbit, s tht, t the end f the ltus rectum (i.e. t the psitin f the cllisin) the helicentric distnce is q. Then, using the frmul fr the speed in prblic rbit (equtin 9.5.), we hve where M is the mss f the Sun. GM V, () q semi mjr xis the elliptic rbit f the cmbined bject. Then, using the frmul fr the speed in n elliptic rbit (equtin 9.5.), we hve Frm equtins () nd () we btin V V GM. () ( + k) q, ( + k) q q q k( + k) r. () ( + k) Als, the cllisin results in n chnge in the ngulr mmentum f the system, s tht, using the frmuls fr the ngulr mmentum per unit mss in prblic nd elliptic rbits (equtins 9.5.7 nd 9.5.8), we btin q ( + k) ( e ). (4) Equtins () nd (4) give us (by elimintin f )
e 4 + 4k + 4k + k, (5) 4 ( + k) r e 4e k + (4 6e ) k + 4( e ) k + ( e ) k 0. (6) () We re tld tht e 0.8, which results in k.06098, s the mss f the bject ws.06098m. (b) Befre embrking n clcultin, here re sme preliminry qulittive thughts. Wht hppens if the mss f the bject is very smll, s tht k is clse t zer? In tht cse the rbit f the cmet is brely chnged, nd the eccentricity f the rbit remins e. Wht hppens if the bject is very much mre thn tht f the cmet? In tht cse the bject brely ntices the impct f the cmet, nd it (nd the cmet, which by nw is stuck t it) flls stright twrds the Sun in stright line, gin with e. S, since e is in the extreme cses f very mssive bject nd very light bject, it my be deduced tht, fr bdy f mss cmprble with tht f the cmet, the eccentricity f its rbit will be less thn, nd tht, fr sme prticulr vlue f its mss, the eccentricity f the rbit will g thrugh minimum. Indeed if we plt e : k frm equtin (5), it lks like this: 4 0.95 0.9 e 0.85 0.8 0.75 0.7 0 0.5.5.5 k Nw fr sme clcultin. We hve t use equtin (5), t find fr wht vlue f k is e lest. Slightly tedius, but n setting the derivtive t zer I find
4 + k 4k 8k 5k k 0, (7) This hs nly ne psitive rel rt, which I fund numericlly, by Newtn-Rphsn itertin, t be 0.444, which lked s thugh it might be. I checked by substituting bck int equtin (7) nd fund tht indeed relly is slutin. It ws bit tedius t d this. Hwever, if yu re better t lgebr thn I m, yu might be ble t fctr equtin (7) int ( + k )( k k )( + k + k ) 0, (8) which, indeed, hs nly ne psitive rel rt, nmely k. Then, n substitutin f this vlue int equtin (5), yu btin, fter sme tedium, e / 0.707. 5
4. α θ 90ºθ α+θ A S F B R θ θ F
5 The trjectry is prt f n ellipse, drwn in full bve. The energy f n elliptic rbit depends n its semi mjr xis, s the lest-energy rbit is the ne tht psses thrugh bth cities nd hs the lest semi mjr xis, s the determining the size nd shpe f the ellipse is prblem ne f pure gemetry. Nte tht ne fcus, F f the ellipse is t the centre f Erth. Let F be the ther (empty) fcus. Then the semimjr xis f the ellipse is ( R + S), nd this is lest when AF is perpendiculr t the mjr xis f the ellipse, s drwn. But semi mjr xis f the lest-energy ellipse is R( + sin θ ). S Rsin θ, s the I hve mrked in sme ngles ner t city A, frm which I hpe it will be greed tht ( α + θ) + 90 θ 80, nd s we find tht the lunch ngle fr the lest-energy rbit is α 45 θ. The lunch speed V 0 is the speed where r R, s, using equtin 9.5. fr the speed in n elliptic rbit, we btin V G M R G M R R( + sin θ) GM + sin θ. R sin θ Since we knw the semi mjr xis, we cn prbbly find the perid. The semi mjr xis is R( + sin θ) 45.75 km. Using equtin 9.6., we see tht the perid is P π / GM.0668 0 s. The ngle 80º θ is the true nmly v t deprture frm A, nd the ngle 80º + θ is the true nmly v t rrivl t B. If we cn find the men nmly t these tw pints, we shuld be ble t find the time tken t get frm A t B. We ll need t use Kepler s equtin, s we ll need t knw the eccentricity e f the ellipse. The distnce between the fci is e R cs θ, frm which we find tht e 0.67 070
6 T find the tw eccentric nmlies, E l nd E we need ne r mre f equtins..7d-g. Just mke sure tht yu hve the crrect qudrnt. I mke them E 9. 577 E 0. 46 Nw, Kepler s equtin (equtin 9.6.5), M I mke them E esin E, gives us the men nmlies. M 0. 48 M 58. 568 The difference is 57º.6. The perid f the entire rbit is.0668 0 s, s the 57.6 durtin f the jurney is.0668 0 s. minutes. 60 One lst questin: Wht is the mximum height bve Erth reched by the vehicle? Since this is reltively esy questin, I ll leve it t the reder.
7. α C A R T β P F Q F B PQ is the mjr xis f the ellipse, f length. The distnce BC is ls equl t. F l F is the distnce between the fci, which is e. I m ging t try t find the verticl distnce between P nd Q. I.e., the height f P bve Q. I.e. the distnce QR. The lengths QB, QF, PA, PF, CT re ll equl t ( e). The distnce QT BC BQ CT ( e) ( e) e. Hence the distnce QR is e cs α. But it is ls equl t csβ. csβ Hence e. cs α
8 I hve drwn this fr the cse β > α (ellipse). Yu might ls shw tht yu get the sme result fr β α (prbl) nd fr β < α (hyperbl). If the cne is cylinder ( α 0), the eccentricity f the ellipticl crss sectin is cs β. Here is n lterntive slutin, fr which I m indebted t Pl Achinty f Indi: z Z β d x α O 90º β X The equtin t the cne, referred t the xes OXYZ (the Y-xis directed wy frm the reder) is X + Y Z tn α 0
9 We nw refer t secnd set f xes, Oxyz, s shwn. Crdintes in the tw systems re relted by X Y Z xsin β y x csβ + z csβ z sin β The equtin t the cne referred t this system is therefre ( x sin β z csβ) + y ( x csβ + z sin β) tn α 0 The equtin t the plne is z d Thus the equtin t intersectin between the plne nd the cne is ( x sin β d csβ) + y ( x csβ + d sin β) tn α 0, which, n mking use f trignmetric identities, cn be written x ( cs βsec α) x( d sin βsec α) + y + d ( sin βsec α) 0 This cn be cst in the frm ( x x0 ) y + b, which shws tht the plne sectin f the cne (with β > α) is n ellipse. It my require sme effrt t btin the cnstnts, b nd x 0 explicitly in terms f α, β nd d. Hwever, it is nt necessry t d this, becuse it is lmst immeditely evident tht the rti b must equl cs βsec α. And since, fr n ellipse, b ( e ), where e is the eccentricity, it fllws tht csβ e. cs α
0 4. Nttin: Speed f sterid immeditely befre the explsin V. Speed f prt tht mves in circulr rbit immeditely fter the explsin V. Speed f ther prt immeditely fter the explsin V. Semi mjr xis f the elliptic rbit f this prt Mss f Sun M. Grvittinl cnstnt G. Sme relevnt equtins: V G M r V GM r V G M r mv + mv (m) V r These shuld be enugh t shw tht. 4r 5