1、1 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 2 1-1 3 m, t, q A r A e r A A e A A A r r r = = 4 : A r B r C r B A C r r r + = A r B r C r A r B r C r D r C B A D r r r r + + = 1) 5 ) ( | or ), ( | | | | | 0 0 = = c A B c A B cA B B A c r r r r r r 2) A r B r C r ) ( B A B A C r r r r r + = =6 3) 4) A r B r cos AB
2、 B A = r r C B A r r r = sin AB C = r 0 : , : |, 2 = = = = B A B A AB B A B A A A A A B B A r r r r r r r r r r r r r r A r B r C r *7 AB B A B A B A B A A B B A = = = | | :, : |, r r r r r r r r r r r r 0 5) k A j A i A A z y x r r r r + + = x i A x : r x A x : x A z A y A x y z A r i r j r k r8 k
3、B A j B A i B A B A z z y y x x r r r r r ) ( ) ( ) ( + + = z z y y x x B A B A B A B A + + = r r k B A B A j B A B A i B A B A B A x y y x z x x z y z z y r r r r r ) ( ) ( ) ( + + = k B j B i B B z y x r r r r + + = 2 2 2 z y x A A A A + + = x A z A y A x y z A r 1 2 2 2 = + + = = = cos cos cos A
4、A cos , A A cos , cos z y A A x k A j A i A A z y x r r r r + + =9 ) ( x f y = ) , ( ), ( ) ( t r B t v t A A r r r r r = t t t + ) ( ) ( t t A t A + r r ) ( ) ( t A t t A A r r r + = : A r ) (t A r ) ( t t A + r A r 2 A r 1 A r 2 1 A A A r r r + = t A A v r v r B t t + B v r10 t t A ) ( r t A t t A
5、 t t A dt A d t t = + = r r r r lim lim ) ( ) ( 0 0 A d A t dt A d r r r , 0 A dt A d t A r r r ) ( k t A j t A i t A t A z y x r r r r ) ( ) ( ) ( ) ( + + = k dt dA j dt dA i dt dA dt A d z y x r r r r + + = ) (t A r ) ( t t A + r A r 11 11 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 12 12 1-2 * ) ) ( ), ( ( t v
6、 t r r r 13 13 : (frame of reference)14 14 . 1 . 2 R 6400km 1.5 10 8 km (particle) 15 15 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 16 16 1-3 ( )17 17 (Position Vector) 18 18 19 19 20 20 P 21 21 op r = r r r p22 22 r r x y z o p r z r y r x z y x r r = = = + + = = cos cos cos 2 2 2 r k z j y i x r r r r r + + =
7、1) (Cartesian coordinates) 23 23 2 (polar coordinates) : r e r r e r r r r = ) ( : e e Note r r = e r p o r r e r ) ( r r e e r r = : e r (?)24 24 3) (Natural coordinates) ) ( : O S : t e r : n e r t e r n e r o S25 25 ) (t r r r r = (Equation of kinematics) 26 26 k t z j t y i t x t r r r r r ) ( )
8、 ( ) ( ) ( + + = ) ( ) ( ) ( t e t r t r r r r = ) ( ) ( ) ( = = = t z z t y y t x x or) ( ) ( = = t t r r or 1) Cartesian coordinates 2 (polar coordinates)27 27 Q ) (t S S = 3) (Natural coordinates) 28 28 ) , ( c y x f = c r f = ) , ( 2 2 2 sin cos r y x t r y t r x = + = = equation of trajectory )
9、 , ( c y x f = c r f = ) , ( ) , ( c y x f = c r f = ) , ( 29 29 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 30 30 1-4 . . ( )31 31 . ( Displacement) A B o t t + t r r 1 r r 2 r r s 1 2 r r r r r r = A B C 1 t 2 1 t t + 2 t BC AB AC + = k z j y i x r k z j y i x r r r r r r r r r + + = + + = x y z32 32 (distance)
10、 s ) 1 r s r ) 2 r s t r = 0 r d ds r = A B r r r r r = ) 3 r r r r r s A r r B r r ? r d dr r r 33 33 . (velocity) t r v = r r t rr r r dt t r d t r t v t ) ( lim ) ( r r r = = 0 r t r 0 A B r r t t t + A B r r B A v r (average velocity) (instantaneous velocity)34 34 ( speed): t r dt r d v v t = =
11、= r r r 0 lim35 35 k z j y i x r r r r r + + = k v j v i v k dt dz j dt dy i dt dx dt r d t v z y x r r r r r r r r + + = + + = =) ( dt dz v dt dy v dt dx v z y x = = = , , z y x v v v v 2 2 2 + + = 1) Cartesian coordinates36 36 ) ( ) ( ) ( t e t r t r r r r = dt e d r e dt dr e r dt d dt r d v r r
12、r r r r r r + = = = ) ( ) ( ) ( t e t t e e r r r r r r + = e e r r r / : 0 = = r r e e r r d e e d r r r = e r ) ( t t e r + r ) (t e r r ) (t e r r r e r 1 p 2 p ct t r r = = ) ( ) ( 2(polar coordinates)37 37 e dt d r e dt dr v r r r r + = dt dr v r = dt d r v = e v e v v r r r r r + = dt d = e r
13、r e r v r e dt d e t t e dt e d t r t r r r r r = = = 0 0 lim lim r v = 38 38 3) (Natural coordinates) ) (t s s = t t e dt ds e v v r r r = = t e r v r r s t t r Q 0 0 = lim lim dt ds dt r d v = = | | r39 39 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 40 40 1-5 1. 2. 41 41 A B v v v r r r = t v a r r = 2 2 0dt r
14、d dt v d t v a t r r r r = = = lim t A A v r a r v r B t t + B v r (average acceleration) (instantaneous Acceleration)42 42 k z j y i x r r r r r + + = k v j v i v v z y x r r r r + + = k dt dv j dt dv i dt dv dt v d a z y x r r r r r + + = = k dt z d j dt y d i dt x d dt r d r r r r 2 2 2 2 2 2 2 2
15、 + + = = 2 2 2 2 2 2 , , dt z d dt dv a dt y d dt dv a dt x d dt dv a z z y y x x = = = = = = 1. Cartesian coordinates43 43 n e r 2. Natural coordinates t e dt ds v r r = dt e d dt ds e dt s d dt v d a t t r r r r + = = 2 2 n t e e r r / : 0 = = t t e e r r d e e d n t r r = ) ( ) ( t e t t e e t t
16、t r r r + = t e r s o 1 p 2 p ) (t e t r ) ( t t e t + r44 44 n n t t t t e dt d e t t e dt e d r r r r = = = 0 0 lim lim dt ds ds d dt d = s ds d s = 0 lim Determined by trajectory ) ( ), ( : t t e t e t t + r r 2 1 : p p s 0 , 0 s s ds d k s = = 0 lim o 1 p 2 p ) (t e t r ) ( t t e t + r t e r s R
17、 k Rd ds / 1 = = 45 45 2 / 3 2 2 2 ) ( 1 + = dx dy dx y d k k 1 = ) (x f y = (radius of curvature)46 46 n t n t e v e dt dv e dt ds e dt s d r r r r 2 2 2 2 1 1 + = + = ) ( 2 1 v a n = n n t t e a e a a r r r + = ) 0 ( , / , 0 dt dv v v e a a t t t r r dt e d dt ds e dt s d a t t r r r + = 2 2 n t e
18、 dt d dt ds e dt s d r r + = 2 2 2 2 dt s d dt dv a t = = p t e r a r n a t a n e r (normal acceleration) (tangential acceleration)47 47 * R dt d R R dt d dt ds v = = = = ) ( R dt d R dt dv a t = = = 2 2 2 ) ( 1 1 R R R v a n = = = R s : dt d = : 2 2 dt d dt d = =48 48 = = 2 2 6 2 t y t x t n a a v
19、v v r r t t , (4) , ) 3 ( , , 2 1 (2) ) 1 ( 2 1 r r r r = = 2 / 6 2 x y = x y 1 v r 2 v r r r 1 r r 2 r r j t i t r r r r ) 2 6 ( 2 2 + = , j i r r r r 4 2 1 + = j i r r r r 2 4 2 = j i r r r r r r r r 6 2 1 2 = = 0 1 2 = = r r r ) / ( 6 2 s m j i t r v r r r r = = j t i dt r d v r r r r 4 2 = =49 4
20、9 j t i dt r d v r r r r 4 2 = = dt dv a t = 2 2 4 1 8 ) 4 1 2 ( t t t dt d + = + = 2 v a n = 1 = = y x y 2 / 6 2 x y = 2 3 2 2 3 2 2 3 2 ) 4 1 ( 1 ) 1 ( 1 ) 1 ( 1 t x y y + = + = + = 2 2 3 2 2 4 1 4 ) 4 1 ( ) 4 1 ( 4 t t t a n + = + + = 2 2 2 a a a t n = + j dt v d a r r r 4 = = 16 2 = a 2 2 2 4 1
21、4 t a a a t n + = = , ) ( j t i t r r r r 2 2 6 2 + = 1 250 50 g r n a r v r p P cos g a n = cos g v a v n 2 2 = = 2 v given51 51 R P , O A P r A a r r = r r dt dv dt ds , 0 30 cos 2R r = RA a 2 3 30 sin 0 = = R v RA a a n 2 0 2 3 30 cos = = = dt ds R A v = = 2 3 0 30 o r r P n a r t a r a r v r 2R
22、t a dt dv = s 52 52 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 53 53 1-6 ) ( ), ( ), ( ) 1 t a t v t r r r r r r = 2 2 dt r d a dt r d v r r r r = = , ) ( ) ( ), ( ) 2 t r t a t v r r r :54 54 5: 1 , 1 0 0 = = y x ) ( , ) ( t r j i r j t i t v r r r r r r r + = + = 0 2 2 0 v r x y 0 r r j t i t j t y i t x t r r
23、 r r r r ) 1 ( ) 2 1 ( ) ( ) ( ) ( 2 + + + = + = 2 0 0 1 ) ( ) ( t dt t v y t y t y + = + = + = + = t x t dt t v x t x 0 0 2 1 ) ( ) (55 55 6: ) (, 0 0 t r v r and a r r r r t a v dt a v t v t r r r r r + = + = 0 0 0 ) ( 2 0 0 0 0 2 1 t a t v r dt v r t r t r r r r r r + + = + = ) ( x y 0 r r 0 v r
24、g r56 56 7: v op X O op v r oM r r Mouse v op , r oM , t g a r r = 2 0 0 2 1 t a t v r r r r r r + + = 2 02 10 t g t v r p p r r r + + = 2 02 1 0 t g r r M M r r r + + = Shooting a mouse57 57 M p r r r r = 0 0 0 + = M p r t v r r M p r t v 0 0 = P M v r t 0 0 = 1 2 ? X O op v r oM r r Mouse 2 02 10
25、t g t v r p p r r r + + = 2 02 1 0 t g r r M M r r r + + = M p r v 0 0 r r |58 58 1.5 (P33) 0 0 1 = dt r d dt dr r) ( dt r d v r r = dt v d a r r = ? 0 0 2 = dt v d dt dv r) ( 0 0 3 = dt a d dt da r) ( 59 59 1.7 (P34): t 1 dt v t x 1 0 AB A B x x t x x dx dt dt dx B A = = = 1 0 1 0 t vdt s ds dt dt
26、ds s t = = = 0 0 1 B A r d r A B r r r r = B A r d r s = B A dr A B r r r r = o B r r A r r r r 60 60 * ( ) T=6s, R=3m, t=0 o t=2s = dt r d r = dt dv = B A r d r = B A r d | | r = B A r d | | r A B O 3 3 2 2 2 = = = = = = R AB R v a T R v , , = dt v d r j i r r 2 3 2 + 0 ) sin (cos j i r r 0 0 30 30
27、 3 + 0 30 3 2 6 cos ) sin (cos j i a r r 0 0 30 30 + 61 61 1-1 1-2 * 1-3 1-4 1-6 1-7 1-5 62 62 1-7 * Relative motion 63 63 64 64 65 65 66 66 a v r a a r r v r r a r e v r e a r 67 67 68 68 69 69 r R r + = r r r dt r d dt R d dt r d + = r r r Q dt R d u dt r d v r r r r = = and ( ) v t d r d dt r d d
28、t dt = = = r r r v u v + = r r r ( ) v u v + = r r r or Position Vector: Velocity:70 70 dt v d dt u d dt v d + = r r r a dt u d a + = r r r Acceleration:71 71 v 30 Earth Man Man Air Earth Air + = v v v v v v Earth Man Man Air Earth Air + = v v v v v v 0 v v v v v v 0 v v v v v v = 0 v v v v v v = 0
29、v v v v v v + = 0 v v v v v v + =72 72 v=4m/s 30 o 60 o (2001 ) v A-E =4m/s v B-E =4m/s v R-A 30 o v R-B 60 o 60 o 30 o RE v r j i v r r r 3 2 2 E - R + = E A A R v v v + = v r r E - R v R-E =? E B B R v v + = v r73 73 Assignments 1.61.8, 1.12, 1.1474 74 * 1-7 P 30 A B B u A v B A B r 0 A v B u A B vu A B B A S , A (