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 ( ϕxyz0≤θ ≤πzrrzxv(t) =v(t) =$` "  $ (+( ( "%  ," " ,$ " $95 95;9: 57 :287 ;5 88;952 453 4;:195 81:22 8:8284 8 489 2 55 84814 98:5 92 5;52 85 : :5%J%.IB?er eθdr(t)≡ ṙ(t) .dtv(t)892765;8:$r(ϕ)θv(t):92 ;92 E4822 :5 9P::85 66:22 #2 : 7 ;2;; 8:(6:5 289( "73 :45 5: 8(87 4229;$2 48:5 8:65 $7y(x)9:P: @*r = er r , dr = er dr + eϕrdϕ .553 2486:2 1 8;222 6 6445 8:5 :595 82325 4 2543 695 4951 :2 577;5 482:57 1;5 ; 5%4;2 7 +4555 92 :97 8!8eϕ+. ( % ((% +( #. er0 ≤ ϕ ≤ 2π95% ;5 12:5 48224( :9"92 422 87 ;5H 4:2I!:H92PF ;: 5 :6:2 1:2 :7642 2( ;28; 8 888(4:5 82 422 484 27 146 "92 69:9 ;2 :224882 ":95 # 5 5 :87 9;$2 3:5 415(89 8%J%;PA#B?8( 57 ":5 457 :' 8 52 8P 8 9"4 : 22 5 F" %8 417 @47$"5 92 8 4 ;57 :5 46 "! 9F18 5 :9 8( P::"92 17 482 8 @8FI7 "8! 22&  45 :& 5 "4 6857 87< "153 5 ;55776:5 ' 465 8(89 2 848: *2 /17 8 4 7955% 15 +( ;5$2 :$7rϕxxeϕr = er r, dr = er dr + eθ rdθ + eϕr sin θdϕ.dr(t).dt45 15 46 17 7; ;2 :9 8! 42 !255 42 " 5% 42: 65 65 4 ;: 4:: 85 ;2 525;:57 92 2 145 *;2 :5 6:2257 ;5 45 ) 6 889 28 : ;7 22 : 5 .4281 4;22 4 42 2 2 : ; 4;5% 6:52 ;7 65 46:6:2 5 8: 2 5) 4925 .42 45 9 65:22 827 ;% ;2 :4; ;7 ;:5 : :57 27 :557 57 ;52 ;:5 42 8 8 8: 57 ;59 88$ P+ #B&  * ( % B & BE! & D?+ B- C  # ))B ? ! )  ) & & ) -) CB% # C ?)D ) ? )E G ?) *+C ? )) #  - C  ) )? $ % C$E) ?% + ,v(t)abα(a, b)ab = ab cos αab = ax bx + ay by + az bzi=jv(t) = dr(t)/dtdvdv/dtae x ex = 1 ex ey = 0%J%:-$*&?+*B*C+δij = 1v2%J%9<?*A-$*&?+*B*C+84814"9;22:"589(845:55454"?ez87 7(5:445 85 6:57:9 5%442 6:8 82 :8x 98*482178!"465;5$7557 292 97 55)-87+.8# ;9 95;5 : :5 645 4;2242 ;:57 :;:57 85757 82 : 274927 8 8;92: 2795 8δiji=6 j654":;#5$":85 4 8:92/;*248145"2:5ex ey> +* ) & c) B C ) * +  C ?) *)) D % +- +& G r  A - CB $ * & C) # )  * - @ #C r - & G B & ? ,8781:22;8#7;5584:$ 5 445 4% 498; 9525 " :645 *242 482" ;:57 )57 2289 42; 4:5:5 65292" :#;55 .224 884: #2 *drvx = ẋ, vy = ẏ, vz = ż .ei ej = δij , i, j = x, y, z ,δij = 0v2 = ẋ2 + ẏ 2 + ż 2 .vr = ṙ, vϕ = rϕ̇; v2 = ṙ2 + (rϕ̇)2vr = ṙ, vθ = rθ̇, vϕ = rϕ̇ sin θ; v2 = ṙ2 + r2θ̇2 + r2ϕ̇2 sin2 θ .drr(t)xyb1x(t) = tv0 cos α , y(t) = tv0 sin α − gt2 .2αxv0yRx(t) = R cos ϕ(t) , y(t) = R sin ϕ(t) , | r(t) |= R .r = R(cos ϕ, sin ϕ)"rv=r(t)qvx2 + vy2 = R | ϕ̇ |r(t) = r(t) er (t)ϕ̇(t)er (t)er (t)rv = Rėr , ėr = ϕ̇ (− sin ϕ, cos ϕ) , ėr ⊥ r .:2789r(t)87;:557R%J%3w)D#=E&D+?+vy9(845:55454"vx(t) = v0 cos α , vy (t) = v0 sin α − gt ,28184;952:;:57578g4 7 24 68:5 ;:57;952 495( 57: 5% 15 824 69:52 ;89 "814482 *8 ;9%25 .2% :5;952 657 46: ;5 :9: 2* 4 4;5 857 "4%:5 65 95:55 :22 5$ 8$ 5$2 78:79' 8;:5 41557 ;517 4!528! 8:" :9 9$724 ;5:8*927656:35$&+ 2v02 cos2 α: 9(87 8879;$2 55: 99 95582 %;:57578178!;:5 42 457 17 "! 8: 6:95 :5& :57 8(2 84 ` 8:7& 2 8* : 457 65 17 8!9 8( 419 5577 455 86:2 4482 2 8$7 ' 558 : 87 .: 42 27 57 45 55 6:5% ;86 ;92 :5 45% ! 27 27 6!2 9 ;% 45 22 45 :5 78 7 87 *892 42& 5 4 .+8# 8: 9%& 27 6:9 17 ;:5572 7 53 ;5 "% 92 $7 5 42 "8! 8 "( 6 7 ; 92 9 86+:;55;55 ;5 2 45 1 : :5 8 M -6::8 4 6: ;% ;57 ' $7 :( " 5 :4 *45 :5 9 5 :57 ( ;92 "4 89 *2 65 8:7 495 +482 : 8 :23/ :5 8577 81 8( 6:27 8 4:2 ! 8$7 ) 3 ' 9 53 42 28 7 5'R 5( ;92 5 52 86 + "7 7 4 9 :& 655 418 :5 :5 65 6:52 .8 :5 ; !8574:2 87N' *;: 1 187 57 4:53 1452 8;57 8) ;62 $ 5 $ $ 5 %$ # 5y(x) = x α −53.2712:879;$2:5879;$2:5$ ?8 495 8! ?8( 92 :6:2 ": 4995 *2 4825 42 ( 8:9  442 +8 :2 :86 ;5 ;257 88: :598 *5 +8195 4655 8!7 95  9 :522 18 42 9 27 45  84 89 % 2 tx2 .v = (ẋ, ẏ) = Rϕ̇ (− sin ϕ, cos ϕ)rv = xvx + yvy = 0 .rr(t) = R er (t) , er k r ,r8;5 :52 6:2"2( 88 4 4 " 59857 2*87 :812 9;827 : 88; $7:855 ;9524 : 882m;1559;:5757:9.482:879;$2:5r2r1 = (r1, θ1, ϕ1)5:92:879;$2:659:524274"():55 5%74:5"%27 6955 6652 ;587 :5 1:2 24 :5 4 187 8412 92 6:59;2 2 55:5 82 47 442 84577 82 12:92 9 84:17 878 9 ;92 76:2 22 :5 572 9v = vt + vn , vt = ṙ er , vn = r ėr , vt k ±r , vn ⊥ r .+ .* &( $ 8: P F ( 56 &8F '( F8 " 4H+.

8 @ 5F6 8$F ('! $ $;952 453: :58 (5-*96 86:52 41452 9527 ;4 972+46 .822:94882-952 :87$v = ṙ er + r ėrdr = ṙdt = vt dt , vt ≡ vdϕ =r2 = (r2, θ2, ϕ2)xvndr = ṙdtvndt,rdϕ vn=dtrr192 5 ?; ' 5 *25) : 8(5 2 ' 49;2 ; 7:57 865 ;;;:5 92 6557 818 48 42 6:92(& 17/ +488( (& 4422 17/ 8 8( 4;! 4422 88 8 48 95 6 57 : 24 1 92 482 195 , 53 8" ; ;2 ," :92 ;5 8:& 44/ 7 " 98 :7 >' %7 *248 45 7 :5 42 85 87 42 952 4#5 1;85 42 223 1;2 87 94 55 *52 :( 42 :2231;85 4:3"73 92 357 :2 ,; ;578 37 12 7 55 ?" 5 9 $ $ xvt%J%0KC?#%1.(224 46:52 822 84"27 :6:5 89$75 52 ;45% :577 (9 2 592+ :.2% ;:57 57 :982 452r(t)a = (30; 0)v2 = (20; − 30)v1 = (− 20; 0)v1 = 2, 5:9 4:2 8 :92 42 (2 ,2 ; 45 ; 4 8 : 88 8 57 18 1 :55 " 2:7 87 ;92 577 6:9 9 83 8 84 954 " : ":5 ;:55 895 ,: *482 ;57 8 ;:582 *27 68: ;57 7 655 3 :57 3:57 8 5 :57872589*5*6;4325 2 + ; :5 " % "4,: 765 ,:95 :557 85( 14 2%3 ;8 8588 ' 42 ;5 87"4 45% 8% 657 4:958# 12 88 * 95 15 : 45 22;:57 9) 695 67 8:5( ;84 57 "4 44557 5 2 6:2 4 48 8 ; 8(8 5 * 42 17 4 695 82 557 6442 827& 8; :22( 45 4:#52 48# 927 ;953 7 4: ;; 6: 7 2 92 8;$ "$ 52 9$ + 4$ 4$ 598(785665:12245%5779+.2%;:57578er (t)AAC = 12 CB = lg = 10BAv2 = 42Bvtvndtdr = vdt = ṙdt er + r ėr dt .dtr dr=.r dtABb = (0; 30);8 ; ` 8:765 :557 i 95346 45:9 17 56482 "8! 2 42+8865 5 89(345%*2 864 8( 7 :#9522 885 4 4857:77 ;7 2;85: 65;:57244 2824" 48 2v(t) = v(0) +0r(t) = r(0) +0a(t) ≈ const = gZtg ≈ 9, 8v(t)a dt = v(0) + a t ,an = −Rϕ̇2(cos ϕ, sin ϕ) = −ϕ̇2 r, an = Rϕ̇2 =1v(t) dt = r(0) + v(0) t + a t2 .245% 482̈ ;92 2 8@4 4( .2 4648 :2 "59595 57 ;:7 97 M5 + 86 49255 6:% 5 2*84;:7:75P 5 ;:557 92 87( :55 ;7( 56F :5 8955 8 69887 9 1 455 55+ 57 8 5 5 65 @@5 42 *4:59 ;92 :92 955 84 ;92 87 :5 : ; :5 P: :5 $7482 8 ; 92 7 92P9;( :4 ;8(244 .

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