1.1 ft t 2 ft = t 2 ft+ t = t+ t d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

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1 A c 2008 by Kuniaki Nakamitsu t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1 t 2 d t

2 1.1 ft t 2 ft = t 2 ft+ t = t+ t d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 = 2t. t 0 dft f t dft df t d c = 0 c, 1.2 d t n = ntn 1, 1.3 d sin at = a cos at, 1.4 d cos at = a sin at, 1.5 d e at = ae at, 1.6 d ft + gt = dft d cft = c dft d ftgt = d ft 2 + dgt, 1.7 c, 1.8 gt + ft d gt. 1.9

3 dft = 0 ft = x = 3 cos 2t 4t 2 + 5t + 6 dx dx = d 3 cos 2t 4t2 + 5t + 6 d cos 2t = 3 4 d t2 + 5d t + d 6 1.7, 1.8 = 3 2 sin 2t 4 2t , 1.3, 1.2 = 6 sin 2t 8t + 5. x = t 2 sin t dx dx = d t2 sin t = d t2 d sin t sin t + t2 1.9 = 2t sin t + t 2 cos t. 1.3, 1.4 x fx t dfx = dfx dx 1.11 dx 1.11 b a = b c c a 3

4 x = t dx4 dx 4 = dx4 dx 1.11 dx = dx4 dt x = t dx = 4x 3 2t 1.3, 1.2 = 4t t x = t = 8tt d sint2 + 1 sint sin x, x = t d2 ft 2 d sint = d sin x t = x = d sin x dx 1.11 dx = d sin x dt x = t dx = cos x 2t 1.4, 1.3 = 2t cost x = t ft d 2 ft 2 = d dft d2 d d 2 C 1 C 2 x x = C 1 cos at + C 2 sin at 1.13 d 2 x 2 = a2 x

5 dx = C 1a sin at + C 2 a cos at d 2 x 2 = d dx 1.12 = d C 1a sin at + C 2 a cos at 1.15 = C 1 a 2 cos at C 2 a 2 sin at = a 2 C 1 cos at + C 2 sin at = a 2 x x 1.13 d 2 x 2 = a2 x x = C 1 cos at + C 2 sin at C 1, C t 3 3t 2 3t 2 t 3 0 t 3 + C C 3t 2 3t 2 t 3 + C C ft ft ft 3t 2 3t 2 3t 2 t 3 + C C 3t 2 = t 3 + C C 5

6 sin 2t sin 2t a = 2 d cos 2t = 2 sin 2t 1 2 d 1 2 cos 2t = sin 2t sin 2t 1 cos 2t 2 0 sin 2t 1 cos 2t + C C 2 sin 2t = 1 cos 2t + C 2 C. ft = + C C 1.18 C C c = ct + C c, 1.19 t n = tn+1 + C, 1.20 n + 1 sin at = 1 cos at + C, 1.21 a cos at = 1 sin at + C, 1.22 a e at = 1 a eat + C,

7 ft + gt = ft + gt, 1.24 cft = c ft c c = 0 0 = C sin 2t 3t C C C 4 sin 2t 3t = 4 sin 2t 3 t , 1.25 = , 1.20, 1.19 C 2 cos 2t 31 3 t3 + 2t + C C = 2 cos 2t t 3 + 2t + C b a ft ft C t = b t = a 2 t 2 t 2 = 1 3 t3 + C C t3 t = 2 t = 1 2 [ ] 1 2 t 2 = 3 t3 = =

8 [ ] b a t = b t = a b ft [a, b] ft [ ] b a a b ft = [F t] b a 1.27 a F t ft C dft dft C 1.27 ft+ b a dft = [ft]b a = fb fa. b a dft = fb fa 1.28 b a ft ft t t = a t = b t 2 m kg s = 8

9 cm g s kg g 100 m 10 s s = V = V = 100 m 10 s = m s = 10m s. t ft dft = ft t. 2.1 t s ft m dft = ft t = m s dft = 1.1 = ft t ft + t ft t ft = t. 9

10 3 3.1 x t t time x x t x t x = t 2 t x t 2 x x v x t v = dx 3.1 v velocity a v t a = dv 3.2 a acceleration x t x = t 3 v a v = dx

11 = d t3 = 3t 2, x = t 3 a = dv = d 3t2 = 6t. 3.2 v = 3t 2 x x a a = d2 x x t x = cos t a a = d2 x = d dx 1.12 = d d cos t x = cos t = d sin t = cos t. 11

12 3.2 3 cm = 3.4 v a 2.1 v = dx = x t dx 3.1 v = 2.1 x t = 3.5 a a = dv = v t dv 3.2 a = 2.1. =. 3.5 = t 1 12

13 t s s = x = sin 2t cm v a v a v = dx 3.1 d sin 2t = = 2 cos 2t, a = dv 3.2 d 2 cos 2t = v = 2 cos 2t = 4 sin 2t. s cm v = = cm s, a = 2 = cm s v = 2 cos 2t cm/s a = 4 sin 2t cm/s x a v v = a

14 v C 1.18 C v x x = v 3.8 x C ft ft a a = cos 2t v t = 0 v 0 v = = a 3.7 cos 2t a = cos 2t = 1 sin 2t + C C v = 1 sin 2t + C 2 C t = 0 v v 0 v 0 = 1 sin 0 + C. 2 14

15 sin 0 = 0 C = v 0 v = 1 2 sin 2t + v 0. v v = 2t x t = 1 x 0 x = = v 2t 3.8 v = 2t = t 2 + C. C 1.20 x = t 2 + C C t = 1 x x 0 x 0 = C C = x 0 1 x = t 2 + x x ma = F 3.9 m a F m mass F force F F F > 0 x. F < 0 x

16 m F 3.9 a m a 3.9 F m d2 x 2 = F m dv = F F = ma 3.9 = m a = = = m kg s 3.13 = kg m/s 2 kg m/s 2 N 10 kg 5 m/s 2 F 16

17 3.9 m = 10 kg a = 5 m/s 2 F = 10 kg 5 m kg m = 50 = 50 N. s2 s 2 F 1, F 2, F 3,... F F = F 1 + F 2 + F x F ma = 0. a a = v v = = a = C 1.26 v v F 0 F = m dv 3.12 = m 0 v dv = 0 = 0. 17

18 0 3.6 x x x x F = mg x = 0 m F F = mg 3.15 g 3.15 F x x

19 ma = mg 3.16 a = g m 3.17 g a g g 9.8 m/s 2 t = 0 x 0 t x t = 0 0 v x v v = a 3.7 = g 3.17 = gt + C. C 1.19 v = gt + C C t = 0 v 0 0 = g 0 + C C = 0 v = gt

20 x x = v 3.8 = gt 3.18 = 1 2 gt2 + C. C 1.20 x = 1 2 gt2 + C C t = 0 x x 0 x 0 = 1 2 g 0 + C C = x 0 x = 1 2 gt2 + x 0. t = 0 v 0 t x 3.18 v = gt + v 0 x = 1 2 gt2 + v 0 t + x x x = 0 F = kx 0 x x 20

21 x x = 0 F x F = kx 3.19 k x > 0 x < 0 F < 0 x F > 0 x k m m d2 x 2 = kx 3.20 x a ma = kx d 2 x 2 = a2 x x = C 1 cos at + C 2 sin at 3.21 C 1 C 2 x d 2 x 2 = a2 x

22 m k t = 0 x 0 v 0 t x x d 2 x = k 2 m x k 3.21 a = m x k k x = C 1 cos m t + C 2 sin m t C 1, C x C 1, C 2 t = 0 x x 0 x 0 = C 1 cos 0 + C 2 sin 0 cos 0 = 1, sin 0 = 0 C 1 = x C v k k k k v = C 1 m sin m t + C 2 m cos m t 3.25 v = dx 3.1 = d k k C 1 cos m t + C 2 sin m t 3.23 k k k k = C 1 m sin m t + C 2 m cos m t 1.4, 1.5 t = t = 0 v v 0 k k v 0 = x 0 m sin 0 + C 2 m cos 0 22

23 sin 0 = 0, cos 0 = , 3.24, 3.26 x = x 0 cos C 2 = m k v k m k m t + k v 0 sin t m sin t cos t t 2π sint + 2π = sin t, cost + 2π = cos t sin t cos t 2π ft t T ft + T = ft T ft 3.27 T 3.27 f 1 t = sin k m t, f 2t = cos k m t f 1 t f 2 t sin t 2π sint + 2π = sin t t t sin k m t + 2π k = sin m t. k m t 23

24 sin k m k m t + 2π m k = sin k m t. f 1 t t t + 2π m k f 1 t + 2π m k = f 1t. f 1 t T = 2π m k sin cos f 2 t T 3.27 T = 2π m k 3.8 x p m v p = mv 3.28 F 0 d p = m dv 3.28 = F 3.12 = 0 F = 0. p 1.10 T T = 1 2 mv

25 m v T T 3.28 v = p/m 3.29 T p T = p2 2m 3.30 F dt = d dt = F v mv2 = 1 v2 md 2 = 1 2 mdvv = 1 dv 2 m v + v dv 1.9 = 1 2 m 2dv v = m dv v = F v dt = d 1 2 mv2 = 1 2 mdv2 25

26 = 1 dv 2 mdv x v dv = 1 dv m 2v 2 = m dv v = F v 3.12 T dt = F v 3.31 = 0 F = 0 T V V = mgx 3.32 x E E = 1 2 mv2 + mgx de = d 1 2 mv2 + mgx = d 1 2 mv2 + d mgx = F v + mg dx = F v + mgv 3.1 = F v F v 3.15 = 0. 26

27 E 1.10 x 0 v 0 v 3.33 E = 1 2 mv2 + mgx 3.34 x, v E x 0 v 0 E = 1 2 mv2 0 + mgx x 0 E = 1 2 mv2 + mg 0 = 1 2 mv v E mv2 = 1 2 mv2 0 + mgx 0 m 2 v 2 = v gx 0. v = ± v gx 0. v v = v gx 0. x 0 v 0 x 27

28 x = v2 0 2g + x V V = 1 2 kx E E = 1 2 mv kx d 1 2 kx = 1 2 k d x2 = 1 2 k dxx = 1 d x 2 k x + xd x = 1 2 k 2xd x = kxv 3.1 = F v de = d 1 2 mv kx mv2 2 kx2 = d + d = F v F v = 0. 28

29 E 1.10 d 1 2 kx d 1 2 kx2 = 1 2 k dx2 = 1 2 k dx2 dx dx = 1 2 k 2xdx = kxv = F v x 0 v 0 x 3.38 E = 1 2 mv kx x, v E x 0 v 0 E = 1 2 mv kx E mv kx2 = 1 2 mv kx2 0 x 0 v 0 x kx2 = 1 2 mv kx

30 x 2 = m k v2 0 + x 2 0. x = ± m k v2 0 + x 2 0. m m x k v2 0 + x 2 0 k v2 0 + x 2 0 x m k v2 0 + x 2 0 x 0 v 0 v k m x mv2 = 1 2 mv2 = 2 2 = 3.5 = m kg s 3.41 kg m 2 /s 2 kg m 2 /s 2 J 3.4 N J = N m

31 F = mg 3.32 V = mgx V x v t x dv dx = d mgx dx = mg = F F = mg, 3.42 F = kx 3.37 V = 1 2 kx2 V x x dv dx = d dx 1 2 kx2 = kx = F F = kx 3.43 F V x 3.42, 3.43 F = dv dx 3.44 F F x V 3.44 V F V 31

32 V C V = + C C 3.45 F F = sin x 3.46 F V V = cos x + C C 3.47 dv dx = dcos x + C dx = sin x = F 3.46 V C 3.44 x = 0 V C 3.46 V x = V 3.47 V = cos x + C. x = 0 V 0 C = 1 cos 0 = 1 V x = 0 V = cos x 1 32

33 F 3.44 V E E = T + V 3.48 T E de = dt + dv = F v + dv 3.31 = F v + dv dx 1.11 dx = F v F v = 0. E 1.10 F F = x 3 E F x = 0 V = 1 4 x4 + C C 3.49 V x dv dx = d 1 dx 4 x4 + C = x 3 = F 3.44 V F x = C = 0 V = 1 4 x4 33

34 3.48 E = 1 2 mv x x = a x = b F W = b a F dx 3.50 W work F m h W g W mg 3.15 F = mg 0 h mg W = = h 0 h 0 F dx mgdx 3.50 = [mgx] h t x = mgh mg 0 = mgh W = mgh 3.32 h mgh 34

35 mg 0 = 0 mgh W k x 0 W W kx 3.19 F = kx 0 x 0 W = = = x0 0 x0 0 F dx kxdx [ 1 2 kx2 ] x0 = 1 2 kx k 02 = 1 2 kx t x W = 1 2 kx2 0 x kx k 02 = kx2 0 W 3.11 x A B A B F A B A F B F B = F A A B 35

36 3.28 A B p A, p B p A, p B p A + p B = p A + p B 3.28 m A v A + m B v B = m A v A + m B v B 3.52 m A, m B A,B v A, v B A,B v A, v B A,B m Av A m Bv B2 = 1 2 m Av 2 A m 2 Bv B t A, B v A t, v B t t A, B F A t, F B t 3.12 A B d v A t d v B t m A = F A t, m B = F B t A B m A v A t + m B v B t t d m d v A t d v B t Av A t + m B v B t = m A + m B = F A t + F B t

37 t F A t + F B t = A B t A, B F A t, F B t d m Av A t + m B v B t = 0 A,B m A v A t + m B v B t 1.10 A B 3.52 m A v B B A B A B v A v B A B m A v A mv A + mv B = mv B mv 2 A mv 2 B = 1 2 mv2 B m v B = v A + v B m/2 v A2 + v B 2 = v A + v B 2 v Av B = 0 37

38 v A = v B = A v B = v B B B A B v A = v B A v B A B A A A B A + B A B 38

39 A B B A + B A A + B A B A + B B A A k k A A k k = 2 2 k A ka k ka k A 1 A A A A B A + B A A A = A + A A 0 0 A A A A A A 39

40 A B A B A B = A B cos θ 4.1 θ A B θ B θ A A B A B = 0 θ = 90 cos θ = A 2 B 3 A B 2π 3 A B A B = 2 3 cos 2π = cos 2π 3 = cos 120 = 1 2 = 3. A A A = A A 4.2 A A A A = A A cos = A 2 cos 0 = 1. 40

41 4.2 xy A A x A x y A y A = Ax A y. 4.3 A x, A y A x y 4.3 y A A y A x x = A 0 0 A x A y 0 A x 2 y 3 A 2 = 3 41

42 2 y 3 A x A, B A = Ax A y, B = Bx B y A + B Ax + B x = 4.5 A y + B y ka kax = k 4.6 ka y Ax A y + Bx B y Ax kax k = A y ka y Ax + B x = A y + B y, A = 5 3 B = A 3 B 2A 3B 5 4 = = =

43 4.5 A x A x B x B x A B A + B x A x + B x y 4.6 k > 0 A x A x A k A x ka x y y A y + B y B y B A + B A y A x 0 B x A x A x + B x ka y 4.5, Ax + 0 A + 0 = = A y + 0 k 0 0 k 0 = =. k 0 0 A + 0 = A, y A y ka A A x ka x Ax A y k 0 = 0 4.9, x A B A = Ax A y B = Bx B y A A A B A B A = A 2 x + A 2 y,

44 A B = A x B x + A y B y 4.11 Ax A y = A 2 x + A 2 y, 4.12 Ax Bx = A x B x + A y B y 4.13 A y B y A = 4 3 B = 2 5 A A B A = = 25 = 5, A B = = A = A = A 2 x + A 2 y. y y A y A A x x B y B A y θ A x B x A x 4.11 A B θ A x θ A B x θ B A B = A B cos θ

45 = A B cosθ B θ A θ = θ B θ A = A B cos θ B cos θ A + sin θ B sin θ A = A B cosx y = cos x cos y + sin x sin y A x B A + B y A y B A cos θ B = B x / B, cos θ A = A x / A, sin θ B = B y / B, sin θ A = A y / A Bx = A x B x + A y B y. A B = B A 4.14 A + B C = A C + B C 4.15 A B + C = A B + A C 4.16 ka B = k A B 4.17 A kb = k A B 4.18 A, B, C A = Ax A y, B = Bx B y, C = Cx C y A B = A x B x + A y B y 4.11 = B x A x + B y A y = B A A + B = Ax + B x A y + B y, 45

46 A + B C = A x + B x C x + A y + B y C y 4.11 = A x C x + A y C y + B x C x + B y C y = A C + B C A B + C = B + C A 4.14 = B A + C A 4.15 = A B + A C ka kax = ka y ka B = ka x B x + ka y B y 4.11 = ka x B y + A y B y = k A B A kb = kb A 4.14 = k B A 4.17 = k A B ka = k A 4.19 A + B A + B k k 46

47 4.19 A ka = A = Ax A y k A = kax ka y ka x 2 + ka y = k 2 A 2 x + A 2 y = k 2 A 2 x + A 2 y = k A 2 x + A 2 y k = k 2 = k A t cos t sin t Ax A = A y d A = d Ax A y = da x da y. da x da y

48 d 2 d 2 A x Ax = 2 2 A y d 2 A y A = t 2 sin t d2 A 2 d 2 A = d2 2 2 = = t 2 sin t d 2 t 2 d sin 2 t 2 sin t d Ax = A y d A = 0 A Ax A y = C1 5.1 da x 0 da y = 0 da x = 0, da y = 0 C 2 C 1, C 2 A x = C 1, A x = C 2 C 1, C

49 6 6.1 xy x y x y y y r x x r 4.3 r r = x y r x, y r t cos t x = cos t y = sin t r = sin t t v r t v = d r 6.1 v v 49

50 a v t a = d v cos t r = v sin 3t a v v = d r 6.1 = d 1 cos t sin 3t d 1 cos t = d sin 3t 5.1 sin t =, cos 3t a = d v 6.2 = d sin t cos 3t d sin t = d 3 cos 3t 5.1 cos t =. 9 sin 3t v v = = sin t cos 3t 2 sin 2 t + 9 cos 2 3t. 6.3,

51 r a a = d2 r r = t + 1 t a a = d2 r 2 = d2 = = t + 1 t d 2 t d 2 t t. 5.2 r v a v r a v a v y a v r x 51

52 6.2 F F 1, F 2, F 3,... F F = F 1 + F 2 + F 3 + F m a = F 6.5 m a F m d2 r 2 = F 6.6 r m d v = F 6.7 v 6.5 F a = 1 m F a v F 52

53 y a = 1 F m v F x 6.3 F F = = F = 0 0 F = m d v = 0. 1/m 4.9 1/m 0 d v = v v v v v 53

54 v = C1 C 2 C 1, C 2 F = m d v = m d = m = m C1 C 2 d C 1 d C 2 = = r m A y r v r sin ωt A r ωt x r cos ωt A t = 0 x x t ωt ω ω A = ωt/t = ω A x r cos ωt y r sin ωt A r r cos ωt r =. 6.9 r sin ωt 54

55 A v a v = a = rω sin ωt rω cos ωt rω 2 cos ωt rω 2 sin ωt, v = d r 6.1 = d r cos ωt 6.9 r sin ωt d r cos ωt = d r sin ωt 5.1 rω sin ωt = 1.5, 1.4, rω cos ωt a = d v 6.2 = d rω sin ωt rω cos ωt d rω sin ωt = d rω cos ωt rω 2 cos ωt = rω 2 sin ωt , 1.5. A 6.5 A F F = m a

56 rω 2 cos ωt = m rω 2 sin ωt r cos ωt = mω 2 r sin ωt r cos ωt 6.9 r sin ωt r F = mω 2 r 6.12 A r A A A F 6.12 r mω 2 r F A A F A 6.12 F A r F = mω 2 r F = mω 2 r 6.12 = mω 2 r 4.19 = mω 2 r cos ωt 2 + r sin ωt = mω 2 r cos 2 ωt + sin 2 ωt = mω 2 r 1 = mω 2 r. A v v v v 56

57 6.10 v = v = rω sin ωt 2 + rω cos ωt 2 = r 2 ω 2 sin 2 ωt + cos 2 ωt = rω sin 2 ωt + cos 2 ωt = rω 1 = rω 6.10, 4.10 ω = v r F = m v2 r A A v 6.14 A t r ωt 6.4 rωt t A v v = rωt t = rω xy p p = m v

58 m v p v p = px p y, v = vx v y px p y = m vx v y p x = mv x, p y = mv y p F 0 d p = m d v 3.28 = F 6.7 = 0 F = p p T T = 1 2 mv2 v 2 = v v 6.17 m v v v v 2 m v v = 2t 1 t 2 58

59 T T = 1 2 mv = 1 2 m v v 6.17 v2 v v = 1 2 m2t2 + 1 t 2 2 v v 4.11 = 1 2 mt4 + 2t = 1 2 mt T 6.15 p T = p2 2m p2 = p p 6.18 p p p v 2 v 2 = v v = 1 m p 1 p 6.15 m = 1 m p 1 p 4.17 m = 1 p p 4.18 m2 = 1 m 2 p2. v p v p v 2 = v v = v 2 x + v 2 y, p 2 = p p = p 2 x + p 2 y T = 1 2 mv2 x + v 2 y,

60 T = 1 2m p2 x + p 2 y 60

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

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