d dx x 2 = 2x d dx x 3 = 3x 2 d dx x n = nx n1
x dx = 1 2 b2 1 2 a2 a b b x 2 dx = 1 a 3 b3 1 3 a3 b x n dx = 1 a n +1 bn +1 1 n +1 an +1
d dx d dx f (x) = 0 f (ax) = a f (ax) lim d dx f (ax) = lim 0 = = = lim 0 a lim 0 a lim (a )0 = af (ax) f (x + ) f (x) f (a(x + )) f (ax) f (ax + a) f (ax) f (ax + a) f (ax) a f (ax + a) f (ax) a
d dx (2x)3 = 2 3 (2x) 2 = 24x 2 d dx (1 + 2x)3 = 2 3 (1+ 2x) 2 = 6(1+2x) 2
360 o = 2 rad 180 o = rad 60 o = 3 rad
sin(x) cos(x) sin 2 (x) + cos 2 (x) =1 sin( 6 ) = 1 2 cos( 4 ) = 1 2 sin( 4 3 ) = sin( + 1 3 ) = sin( 1 3 ) = 3 2
y = ax + bx 1 1 2 y 1 y = cx + dx y 2 2 1 2 = a c b x 1 d x 2 A = a c b d y 1 y 2 = A x 1 x 2
A = a b c d y 1 y 2 = A x 1 x 2 x 1 x 2 = B y 1 y 2 B = u 11 u 12 = u 21 u 22 u 11 = u 12 = d ad bc b ad bc 1 ad bc u 22 = u 21 = d c b a a ad bc c ad bc
A = a c b d dy 1 = adx 1 + bdx 2 by 2 = bcx 1 + bdx 2 y 1 y 2 = A x 1 x 2 dy 1 by 2 = adx 1 bcx 1 d x 1 = ad bc y + b 1 ad bc y 2 y 1 = ax 1 + bx 2 y 2 = cx 1 + dx 2 cy 1 = acx 1 + bcx 2 ay 2 = acx 1 + adx 2 ay 2 cy 1 = adx 2 bcx 2 c x 2 = ad bc y + a 1 ad bc y 2
34000 = 5X + 3Y 42000 = 6X + 4Y 34000 = 5 3 X 42000 6 4 Y 4 3 X = 2 2 34000 Y 6 5 42000 2 2 = 2-1.5 34000-3 2.5 42000 = 2 34000 1.5 42000 3 34000 + 2.5 42000 = 5000 3000
A = a 11 a 12 a 21 a 22 B = b 11 b 12 b 21 b 22 y 1 = A x 1 y 2 z 1 z 2 x 2 = B y 1 y 2 z 1 z 2 = BA x 1 x 2 BA = b 11a 11 + b 12 a 21 b 11 a 12 + b 12 a 22 b 21 a 11 + b 22 a 21 b 21 a 12 + b 22 a 22
y 1 = a 11 x 1 + a 12 x 2 y 2 = a 21 x 1 + a 22 x 2 z 1 = b 11 y 1 + b 12 y 2 z 2 = b 21 y 1 + b 22 y 2 z 1 = b 11 (a 11 x 1 + a 12 x 2 ) + b 12 (a 21 x 1 + a 22 x 2 ) z 2 = b 21 (a 11 x 1 + a 12 x 2 ) + b 22 (a 21 x 1 + a 22 x 2 ) z 1 = (b 11 a 11 + b 12 a 21 )x 1 + (b 11 a 12 + b 12 a 22 )x 2 z 2 = (b 21 a 11 + b 22 a 21 )x 1 + (b 21 a 12 + b 22 a 22 )x 2 z 1 z 2 = b 11 a 11 + b 12 a b a + b a 21 x 11 12 12 22 1 b a + b a b a + b a 21 11 22 21 21 12 22 22 x 2
X 1 X 2 Y 1 Y 2 Z 1 Z 2 Z 1 Z 2 Y 1 Y 2 Z 1 Z 2 = 5 3 X 1 6 4 X 2 = 2 4 Y 1 5 4 = 2 4 5 3 X 1 5 4 6 4 X 2 Y 2 2 5 + 4 6 2 3 + 4 4 = X 1 5 5 + 4 6 5 3 + 4 4 34 22 = X 1 49 31 X 2 X 2
V 0 0 f 0 V 0 = f 0 0 V 2 V 1
(V 0 V 1 )T f 0 T = V 0 V 1 f 0 f '= V 0 = V 0 f V V 0 0 1
T (V 0 + V 2 )T 0 = V 0 T 0 V 0 + V 2 V 0 f = V 0 + V 2 V 0 f 0
f = V + V 0 2 f V V 0 0 1
V 0 = 340m / s V 1 =±19.0827m / s = ±68.70km / h V 2 = 0 340 f = ± 340 m19.0827 f 0 f =1.0595 f + 0 f = 0.9469 f 0 1 12 2 =1.0595
v =1 [m / s] x [m] 5 4 3 2 1 0 0 1 2 3 4 5 t [s]
x [m] 0s 2s v = 4-0 2-0 2s 3s v = 2-4 3-2 3s 5s v = 5-2 5-3 [m / s] = 2 [m / s] [m / s] = -2 [m / s] [m / s] =1.5 [m / s] 5 4 3 2 1 0 0 1 2 3 4 5 t [s]
5 4 3 2 1 0 x [m] 0s 1s v = 2-0 1-0 1s 2s v = 4-2 2-1 2s 3s v = 2-4 3-2 3s 4s v = 3.5-2 4-3 4s 5s v = 5-3.5 5-4 0 1 2 3 4 5 [m / s] = 2 [m / s] [m / s] = 2 [m / s] [m / s] = -2 [m / s] [m / s] =1.5 [m / s] [m / s] =1.5 [m / s] t [s]
1s2s 5 4 3 2 1 0 x [m] v = 3 1 2 1 [m / s] = 3 [m / s] 0 1 2 3 4 5 t[s](t+ )[s] v = t [s] x(t + ) x(t) [m / s] = (t + ) t t v(t) = x(t + ) x(x) lim 0 = dx dt v(t) = dx dt x(t + ) x(t) [m / s]
x(t) = a + a t + a t 2 + a t 3 0 1 2 3 v(t) = d dt x(t) = a + 2a t + 3a t 2 1 2 3
d dx F(x) = f (x) F(x)f(x) f (x) dx = F(b) F(a) a b d dt x(t) = v(t) t 2 v(t) dt = x(t ) x(t ) 2 1 t 1
t=0x=0 t v(t) = b 0 + b 1 t + b 2 t 2 t t x(t) = x(0) + (b + b t + b t 2 ) dt 0 1 2 0 = b 0 t + 1 2 b 1t 2 + 1 3 b 2t 3
d dt v(t)
a(t) = d dt v(t) v(t) = d dt x(t) a(t) = d dt d dt x(t) = d2 dt 2 x(t)
v(t) = c 0 + c 1 t + c 2 t 2 + c 3 t 3 a(t) = d dt v(t) = c + 2c t + 3c t 2 1 2 3 x(t) = b 0 + b 1 t + b 2 t 2 + b 3 t 3 + b 4 t 4 v(t) = b 1 + 2b 2 t + 3b 3 t 2 + 4b 4 t 3 a(t) = 2b 2 + 6b 3 t +12b 4 t 2
a(t) = d dt v(t) t 2 a(t) dt = v(t ) v(t ) 2 1 t 1 v(t) = v(t 1 ) + t t 1 a(t) dt
t=0x=0 t=0 a(t) = b 0 + b 1 t + b 2 t 2 t v(t) = v 0 + (b 0 + b 1 t + b 2 t 2 ) dt 0 = v 0 + b 0 t + 1 2 b 1 t 2 + 1 3 b 2 t 3 x(t) = v 0 t + 1 2 b 0 t 2 + 1 2 1 3 b 1 t 3 + 1 3 1 4 b 2 t 4 = v 0 t + 1 2 b 0t 2 + 1 6 b 1t 3 + 1 12 b 2t 4
m
m F
m 1 m 1 F
m 1 = 1 2 m m 1 1 2 F
mv p = mv
p = mv d dt p = F
F = d dt p t 2 F(t) dt = p(t 2 ) p(t 1 ) t 1 v(t 2 ) = m(t 1 ) m(t 2 ) v(t 1 ) + 1 m(t 2 ) t 2 F(t) dt t 1
p = mv d dt p = F m d dt v(t) = F ma(t) = F a(t) = 1 m F
t=0x=0 t=0 F(t) = c 0 + c 1 t v(t) = 1 m (c 0t + 1 2 c 1t 2 ) x(t) = 1 m (1 2 c 0t 2 + 1 6 c 1t 3 )
m T
n u(t,n 1) u(t,n + 1) u(t,n) C m d2 dt 2 u(t,n) = T sin( n ) T sin( n1 )
x y sin() = y x 2 + y 2 = 1 1+ y x 2 y x y x y x << 1
m d2 dt 2 u(t,n) = T sin( n ) T sin( n1 ) u(t,n + 1) u(t,n) = T T C u(t,n + 1) u(t,n) = T T C u(t, x + C) u(t, x) = T C (x = nc) u(t,n) u(t,n 1) C u(t,n) u(t,n 1) C u(t, x) u(t,x C) C
t u(t,x) = u(t, x) = x Lim 0 Lim 0 u(t +,x) u(t, x) u(t,x + ) u(t, x) f (x,y) = c 1 x + c 2 x 2 + c 3 xy 3 x f (x, y) = c 1 + 2c 2 x + c 3 y 3 y f (x, y) = 3c xy 2 3
C 0 m 2 u(t, x) 2 t u(t, x + C) u(t,x) = TLim C C 0 = T x = TC u(t, x) x Lim 1 C 0 C = TC 2 u(t, x) 2 x x u(t,x C) u(t, x) x u(t, x) u(t, x C) C u(t, x C)
m 2 2 u(t, x) = TC u(t, x) t 2 x 2 2 2 t 2 u(t,x) = T 1 u(t, x) x 2 2 2 t 2 u(t,x) = T u(t, x) x 2 = m C
2 t 2 u(t, x) = v 2 2 x 2 u(t,x)
d dx d dx sin(x) = cos(x) cos(x) = sin(x) u(t, x) = sin(k(x vt)) u(t,x) = kv cos(k(x vt)) t 2 t u(t, x) = 2 (kv)2 sin(k(x vt)) u(t, x) = k cos(k(x vt)) x 2 x u(t,x) = 2 (k)2 sin(k(x vt)) 2 t u(t, x) = v 2 2 2 x u(t,x) 2
sin(k(x vt)) t = 0 x vt t 0 x sin(k(x v vt))
sin(k(x vt)) = sin(kx vkt) = sin(kx t) k = 2 = 2 f f
2f = sin(k(x vt)) = sin(kx vkt) = sin(kx t) k = 2 = 2 f = vk v = f
2 t u(t,x) = T 2 2 x u(t,x) 2 = m C v =± T sin(k(x v vt)) sin(k(x + vt)) v
2 t 2 u 1(t,x) = v 2 2 2 x 2 u 1(t,x) t u 2 2(t,x) = v 2 2 x u 2(t,x) 2 u 3 (t,x) = a 1 u 1 (t,x) + a 2 u 2 (t,x) 2 t 2 u 3(t,x) = a 1 2 = a 1 v 2 2 = v 2 (a 1 2 t 2 u 1(t, x) + a 2 2 x 2 u 1(t,x) + a 2 v 2 2 x 2 u 1(t, x) + a 2 2 x 2 u 2(t,x) x 2 u 2(t,x)) t 2 u 2(t,x) 2 x 2 u 3(t, x) = a 1 2 x 2 u 1(t,x) + a 2 2 x 2 u 2(t,x)
2 t u (t, x) = v 2 2 2 1 x u 2 1 2 t u (t, x) = v 2 2 2 2 x u 2 2 (t, x) (t, x) u 3 (t, x) = a 1 u 1 (t, x) + a 2 u 2 (t, x) 2 t u (t, x) = v 2 2 2 3 x u 2 3 (t, x)
L x = L 2 x = L 2
u(t, L 2 ) = 0 u(t, L 2 ) = 0 u(t,x) = a 1 sin(kx t) + a 2 sin(kx + t) u(t, L 2 ) = a 1 sin( L 2 k t) + a 2 sin( L 2 k + t) = 0 u(t, L 2 ) = a 1 sin( L 2 k t) + a 2 sin( L 2 k + t) = 0
sin(x 1 + x 2 ) = sin x 1 cos x 2 + cos x 1 sin x 2 cos(x 1 + x 2 ) = sin x 1 sin x 2 cos x 1 cos x 2
0 = a 1 sin( L 2 k t) + a 2 sin( L k + t) 2 = a 1 sin( L 2 k)cos t a cos(l 1 2 k)sint a 2 sin( L 2 k)cos t + a 2 cos( L 2 k)sint 0 = a 1 sin( L 2 k t) + a 2 sin( L k + t) 2 = a 1 sin( L 2 k)cos t a 1 cos( L 2 k)sint A + B 2 (a 2 a 1 )cos( Lk 2 )sint = 0 B-A 2 (a 2 + a 1 )sin( Lk 2 )cost = 0 + a 2 sin( L 2 k)cos t + a 2 cos( L 2 k)sint
(a 2 a 1 )cos( Lk 2 ) = 0 (a 2 + a 1 )sin( Lk 2 ) = 0 (a 2 a 1 ) 0 cos( Lk 2 ) = 0 sin(lk 2 ) 0 (a 2 + a 1 ) = 0 cos( Lk 2 ) 0 (a 2 a 1 ) = 0 (a 2 + a 1 ) 0 sin(lk 2 ) = 0
cos( Lk 2 ) = 0 and (a + a ) = 0 2 1 OR sin( Lk 2 ) = 0 and (a a ) = 0 2 1
(a 2 + a 1 ) = 0 u(t, x) =a 1 (sin(kx t) sin(kx + t)) = 2a 1 coskx sint (a 2 a 1 ) = 0 u(t, x) =a 1 (sin(kx t) + sin(kx + t)) = 2a 1 sin kx cost cos( kl 2 ) = 0 kl 2 = 1 2, 3 2, 5 2 k = L, 3 L, 5 L, 7 L sin( kl 2 ) = 0 kl 2 =, 2, 3 k = 2 L, 4 L, 6 L, 8 L
= vk f = 2 = 1 2 (n = 1, 2, 3, 4, L) T k = 1 2L T n
L = 1 2 f T
f 1 = 1 2L f 2 = 1 2L f 2 f 1 = T 2 T 1 T 1 T 2 f 2 f 1 = 1.1 T 2 T 1 = 1.21
E r = 0 B r = 0 E r = t r B r B = μ 0 0 t r E r E r B 0 μ 0
r r r F = QE E Q r A = (A x, A y, A z ) r A = x A x + y A y + z A z r A = ( y A z z A y, z A x x A z, x A y y A x ) U = ( x U, y U, z U)
0 = 107 4c 2 [C2 /Nm 2 ] = 8.85418782 10-12 [C 2 /Nm 2 ] μ 0 = 4 10 7 [N / A 2 ] =1.25663706 10-6 [N / A 2 ] c = 2.99792458 10 8 [m / s]
( r A ) = ( r A ) 2 r A 2 r A = ( 2 x 2 A x + 2 y 2 A x + 2 z 2 A x, 2 x 2 A + 2 y y 2 A + 2 y z 2 A y, 2 x 2 A + 2 z y 2 A + 2 z z 2 A z)
E r = 0 B r = 0 E r = t r B r B = μ 0 0 t r E ( r E ) = t r B r B = μ 0 0 t r E
( r E ) = ( r E ) 2 r E = μ 0 0 2 2 r E = μ 0 0 2 2 r E = μ 0 0 2 t 2 t 2 r E r E t 2 r E
r E = (0, E (x),0) y y E y = 0 z E z = 0 2 x 2 E y = μ 2 0 0 2 t 2 E y = 1 2 μ 0 0 t 2 E y x 2 E y 2 t 2 E y = c 2 2 x 2 E y
D
X X D D
X X = n D n = 1,2,3,4L X = Dsin sin = D n
= 532 nm = 5.32 10-7 m = 650 nm = 6.50 10-7 m
D = 0.02 mm = 2.00 10 5 m = 5.32 10-7 m, D = 0.0266 n =1 sin = 0.0266 = 0.026603 rad =1.5242 o n=2 sin = 0.0532 = 0.053225 rad = 3.0496 o = 6.50 10-7 m, D = 0.0325 n =1 sin = 0.0325 = 0.032506 rad =1.8624 o n=2 sin = 0.0650 = 0.065046 rad = 3.7269 o
V = 3.0 10 4 [m / s] =1.110 5 [km / h]
L L
T 2 = (VT 1 ) 2 + L 2 = (ct 1 ) 2 T 1 = 2L L c + V + c 2 V 2 L c V = 2cL c 2 V 2 T 2 T 1 = 2cL c 2 V 2 = 2L c 2 V 2 2L c 2 V ( 1 2 1 ( V c )2 1) = 2L c 1 ( V c )2 1 1 ( V 1 c )2
V = 0
c = 2.99792458 10 8 [m / s]
t x t v x
t = t t x t = 0 t x
t 0 t x = x vt t = t x t v x vt
x = x vt x = 1 v t 0 1 x t
t=0 xm xm x = 9.8 + 0.1t x = x 5t 0 = 9.8 + 0.1t 5t 4.9t = 9.8 t = 2 x = 0
x = 1 v 1 t 0 1 x t x = 1 v 2 x t 0 1 t x = 1 v 2 1 v 1 t 0 1 0 1 x t = 11 v 0 1 (-v ) v 1 2 1 2 0 1+10 0 (-v 1 ) +11 x t = 1 (v + v ) 1 2 x 0 1 t
v 1 + c c c + v 2 c
x 2 + y 2 + z 2 = (ct) 2 ( x ) 2 + ( y ) 2 + ( z ) 2 = (ct ) 2 y = y z = z x t = u u 11 x 12 u u 21 22 t
y = y z = z x = u 11 u 12 x t u 21 u 22 t x = 0 d dt x = v 0 = u 11 x + u 12 t u 12 u 11 = v x = u 12 u 11 t x v x
x = u 11 u 12 x t u 21 u 22 t x = 0 d dt x = u 12 t t = u 22 t x = v u 12 u 22 = v u 22 = u 11 x = u 12 u 22 t x v x
x = u 11 vu 11 t u 21 u 11 x t x 2 + y 2 + z 2 = (ct) 2 ( x ) 2 + ( y ) 2 + ( z ) 2 = (ct ) 2 ( x ) 2 + ( y ) 2 + ( z ) 2 = (ct ) 2 (u 11 x vu 11 t) 2 + y 2 + z 2 = c 2 (u 21 x + u 11 t) 2 u 2 11 x 2 2vu 2 11 xt + v 2 u 2 11 t 2 + y 2 + z 2 = c 2 u 2 21 x 2 + 2c 2 u 21 u 11 xt + c 2 u 2 11 t 2 (u 2 11 c 2 u 2 21 )x 2 + y 2 + z 2 2(vu 2 11 + c 2 u 21 u 11 )xt = (c 2 u 2 11 v 2 u 2 11 )t 2 (u 2 11 c 2 u 2 21 ) = 1 (1) (c 2 u 2 11 v 2 u 2 11 ) = c 2 (2) (vu 2 11 + c 2 u 21 u 11 ) = 0 (3)
(2) u 11 = c c 2 v = 1 2 1 ( v c )2 (3) u 21 = v c 2 u 11 = ( v c ) c 1 ( v c )2
x t = 1 1 ( v c )2 v 1 ( v c )2 ( v c ) c 1 ( v c )2 1 1 ( v c )2 x t x c t = 1 1 ( v c )2 ( v c ) 1 ( v c )2 ( v c ) 1 ( v c )2 1 1 ( v c )2 x ct x w = x w w = ct w = c t = v c = 1 1 2
w = ct w = ct = v c = 1 1 2 x w = x w x v x
1 = v 1 c, 2 = v 2 c 1 1 = 1, 2 2 = 1 1 1 2 2 x = 1 1 1 x w 1 1 1 w x 2 2 2 x = w 2 2 2 w x 2 2 2 1 1 1 = w 2 2 2 1 1 1 x w = 1 2 (1+ 1 2 ) ( 1 + 2 ) 1 2 ( 1 + 2 ) 1 2 1 2 (1 + 1 2 ) x w
1 1 1 + 2 1 + 1 2 2 1 = (1+ 1 2 ) 2 ( 1 + 2 ) 2 (1 + 1 2 ) 2 (1 + = 1 2 ) (1+ 1 2 ) 2 ( 1 + 2 ) 2 (1 + = 1 2 ) 1 + 2 1 2 + 2 1 2 2 2 1 2 2 2 1 2 (1+ = 1 2 ) 1 + 2 1 2 2 2 2 1 2 (1 + = 1 2 ) (1 2 1 )(1 2 2 ) = 1 2 (1 + 1 2 )
x = 1 2 (1+ 1 2 ) ( 1 + 2 ) 1 2 w ( 1 + 2 ) 1 2 1 2 (1+ 1 2 ) x w = x w = 1 + 2 1 + 1 2 1 = 1 2 = v c v = c v = v 1 + v 2 1+ v 1v 2 c 2
v 2 = c v = v + c 1 1+ v 1 c = c 2 v = c v 1 + c 1+ v 1 c = c
v v = v + v 1 2 1+ v = 1 v 2 c 2 = 1.2c 1.35 = 24 27 c = 8 9 c 0.7c + 0.5c 1+ 0.35
w = ct w = ct = v c = 1 1 2 x w = x w
w = ct w = ct = v c = 1 1 2 x w = x w x v x
w = ct w = ct = v c = 1 1 2 x w = x w x L 0 ( x, t ) L 0 v x
x x = L 2 1 0 x = x 1 x = x 2 ( x, t ) L 0 v x
x (x,t) x = x 1 x = x 2 t = t 0 x 2 x 1 = L
x = x 1 x = x 2 t = t 0 x = w x w t 1 = c x 1 + t 0 t 2 = c x 2 + t 0 x 2 x 1 t 2 t 1
x = w x w x = x 1 x = x 2 t = t 0 x = x c t 1 1 0 x = x c t 2 2 0 x x = (x x ) 2 1 2 1 L < L 0 L = 1 2 L 0 L 0 = L
x = v c = 0.8 L = 0.6 5 = 3.0 1 2 = 1 0.64 = 0.6
x = v c = 0.8 1 2 = 1 0.64 = 0.6 L = 0.6 5 = 3.0 x = x 0
x
v = 0.5c v = 0.5c 0.5c 0.5c v = 1+ 0.5 0.5 = c 1+ 0.25 = 0.8c 1 0.5 0.5 L 0 = 5 5 L 0 = 0.75 = 10 3 m 1 2 = 1 0.8 2 = 0.6 L = 0.6 10 3 = 2 3 m (= 3.46410 m)
L = 1 0.5 0.5 5 = 5 0.75 = 2.5 3 m
x = x 0 t t x t v x
A = a b c d y 1 y 2 = A x 1 x 2 x 1 x 2 = B y 1 y 2 B = u 11 u 12 = u 21 u 22 u 11 = u 12 = d ad bc b ad bc 1 ad bc u 22 = u 21 = d c b a a ad bc c ad bc
w = ct w = ct = v c = 1 1 2 x = w x w = 2 (1 2 ) =1 x = w x w
x = x 0 t t x t v x x = x + ct 0 ct = x 0 + c t
x = x + ct 0 ct = x + ct 0 ct 1 = x 0 + c t 1 ct 2 = x 0 + c t 2 t 2 t 1 = ( t 2 t 1 ) w = ct w = ct t = t = v c = t = t t 2 1 1 1 2 t = t t 2 1
t = t > t 1 t 1 2 x = vt
t = t = t = 500 s 1 t 1 v 2 c 1 t = 1 1 0.64 0.6 t = 5 3 t
(X 1, Y 1, Z 1 ), (X 2, Y 2, Z 2 ), (X 3, Y 3, Z 3 ), (X 4, Y 4, Z 4 ) T 1, T 2, T 3, T 4 (x,y,z) t t
[c(t + t T 1 )] 2 = (X 1 x) 2 + (Y 1 y) 2 + (Z 1 z) 2 [c(t + t T 2 )] 2 = (X 2 x) 2 + (Y 2 y) 2 + (Z 2 z) 2 [c(t + t T 3 )] 2 = (X 3 x) 2 + (Y 3 y) 2 + (Z 3 z) 2 [c(t + t T 4 )] 2 = (X 4 x) 2 + (Y 4 y) 2 + (Z 4 z) 2 T 1, T 2, T 3, T 4 t T 1, t T 2, t T 3, t T 4
L T = 2L c x T L L = 2 L 2 + Tv 2 T = L c 2 Tv x
L = 2 L 2 + Tv 2 2 T = L c T = 2L c ct = 2 L 2 + Tv 2 c 2 T 2 = 4(L 2 + Tv 2 (c 2 v 2 )T 2 = 4L 2 T = 2 2 ) 2L c 2 v = 2L 2 c 1 v c 2 T = T = 1 1 v c 1 2 1 2 T T
c = 2.99792458 10 8 [m / s] sin(kx t) = sin( 2 x 2 f t) = sin( 2 c f (x ct)) sin( 2 c f (x + ct))
sin( 2 c f (x ct)) sin( 2 c f (x + ct)) v v
sin( 2 c f 0( x c t )) sin( 2 c f (x ct)) sin( 2 c f 0 ( x c t )) x = ct x ct = sin( 2 c f ((x ct) (x + ct))) 0 = sin( 2 c f (( + )x ( + )ct)) 0 = sin( 2 c f 0(1 + )(x ct)) f = (1+ ) f 0
f = (1 + ) f 0 = 1 + 1 2 f 0 = (1+ ) 2 (1 )(1 + ) f 0 = 1 + 1 f 0
sin( 2 c f 0( x + c t )) sin( 2 c f 0( x + c t )) sin( 2 c x = ct f (x + ct)) x ct = sin( 2 c f 0((x ct) + (x + ct))) = sin( 2 c f 0(( )x ( )ct)) = sin( 2 c f 0(1 )(x + ct)) f = (1 ) f 0
f = (1 ) f 0 = = = 1 f 1 2 0 (1 ) 2 (1 )(1+ ) f 0 1 1+ f 0
> 0 < 0 f = 1 + 1 f 0 f = V 0 + V 2 V 0 V 1 f 0 = 1+ (V 2 /V 0 ) 1 (V 1 /V 0 ) f 0
10 THz = 10 13 Hz = 0.5 f = 1+ 1 f 0 = 1.5 0.5 f 0 = 3 f 0 = 3 10 13 Hz
f 0 v f 2 f 1
f 1 = 1+ 1 f 0 f 2 = 1 + 1 f 1 = 1+ 1 f 0
if R < 1 S = 1 1 R = 1 + R + R2 + R 3 +
f 2 = 1+ 1 f 0 = (1+ )(1 + + 2 + 3 +) f 0 = (1+ 2 + 2 2 + 2 3 +) f 0 f 2 f 0 = 2( + 2 + 3 +) f 0 if << 1 f 2 f 0 = 2 f 0
v =100 km / h = 27.7778 m / s f 0 =10GHz =10 10 Hz c = 2.99792458 10 8 m/s = v c = 9.2656693 10-8 f 2 f 0 = f = 2 f 0 =1853.13Hz =1.853KHz
y x
( y y 0 )sin + ( x x 0 )cos = 0 y 0 = x 0 = D sin D cos D y ( y y sin + D = D sin )sin + ( x y sin + x cos = x cos D cos )cos = 0 D sin 2 + D cos 2 y = sin cos x x
sin( 2 c f 0( D + ct )) = sin( 2 c f 0( x cos + y sin + c t )) y x
y sin( 2 c f ( 0 x cos + y sin + c t )) x y sin( 2 c f (x cos + y sin + ct)) x
x = x ct ct = x + ct sin( 2 c f ( 0 x cos + y sin + c t )) = sin( 2 c f 0 ((x ct)cos + y sin + (x + ct))) = sin( 2 c f 0(( cos )x + y sin + ( cos + )ct)) = sin( 2 c f 0 ((cos )x + y sin + ( cos + 1)ct)) = sin( 2 c f cos 0(1 cos )( 1 cos x + sin (1 cos ) y + ct))
sin( 2 c f 0( x cos + y sin + ct )) = sin( 2 c f cos 0(1 cos )( 1 cos x + sin( 2 c f (x cos + y sin + ct)) sin (1 cos ) y + ct)) f = (1 cos ) f 0 cos cos = 1 cos sin = sin (1 cos ) cos 2 + sin 2 cos = 1 cos = (cos )2 + (1 2 )sin 2 (1 cos ) 2 2 sin + (1 cos ) = cos2 2 cos + 2 + sin 2 2 sin 2 (1 cos ) 2 = 1 2 cos + 2 2 sin 2 = 1 (1 cos ) 2 2
cos = cos 1 cos sin = sin (1 cos ) =0.99 =0.9 =0.8 =0.5 =0.2 =0
cos = cos 1 cos if = 2 = 90 cos = 0 cos = = v c = 2 sin = v c
V 0 tan = v V 0 v
f = (1 cos ) f 0 cos = cos 1 cos if = 2 cos = 0 cos = 0 f = (1 2 ) f 0 = 1 2 f 0
y f = 1 2 f 0 x
d dt P = F p = m 0 v t t 0 dt 2 d P = t dt F dt t 2 0 2 p(t) p(t ) = dt F 0 2 t t 0
F t p(t) p(t ) = dt F = (t t )F 0 2 0 t 0 m 0 p = m 0 v v(t) v(t 0 ) = v = t t 0 m 0 F v(t) = v(t 0 ) + v
v = v + v 1 2 1 + v 1 v 2 c 2 p(t) p(t 0 ) = p = t F v = tf m 0 v(t) < v(t 0 ) + v
v ( t ) = 0 0 x x t = t t 0 v = v ( t ) v ( t 0 ) p = m 0 v = t F
p = m 0 v = t F v(t 0 ) = v p = tf t = p = t 1 2 t F = m 0 v 1 2 1 2 p = (m + m)(v + v) mv = vm + mv
v = 1+ v + v v v c 2 = (v + v )(1 = v v 2 c 2 = (1 2 ) v v v v c 2 + v + v v v c 2 v v c 2 2 2 ) v v v v c 2 v = 1 1 2 v
vm + mv = m (1 2 ) (3/2) v 0 m = v v m 0 (1 2 ) (3/2) (1 m (1 2 ) (3/2) ) m 0 m v = 1 v m 0(1 2 ) (3/2) (1 m (1 2 ) (3/2) ) m 0 0 dm dv = 1 v m 0(1 2 ) (3/2) (1 m (1 2 ) (3/2) ) m 0
dm dv = 1 v m 0 (1 2 ) (3/2) (1 m m 0 (1 2 ) (3/2) ) if m = m 0 1 2 = m 0 1 (v / c) 2 d dv m = 1 2 m 0 (1 (v / c)2 ) (3/2) (2 v c 2 ) = 1 v m 0(1 (v / c) 2 ) (3/2) 2 (1 m m 0 (1 2 ) (3/2) ) = 1 (1 2 ) = 2
= v c m = m 0 1 2
p = mv = m 0 1 2 v m = m 0 1 2 m 0
m = m 0 = 60 1 2 1 0.64 = 60 0.6 = 100kg
F x E = Fx d dt E = F d dt x = Fv
u 1 = v 1 2 c u = 2 1 2 u 2 u 2 = v 2 c 2 1 2 1 2 = c 2 ( v 2 c 1 v c 1) 2 = c 2
u 1 2 u 2 2 = c 2 d dt (u 2 u 2 ) = 0 1 2 2u 1 ( d dt u 1 ) 2u 2 ( d dt u 2 ) = 0
2u 1 ( d dt u 1 ) 2u 2 ( d dt u 2 ) = 0 v 1 2 d dt m 0 v 1 2 1 d 1 2 dt m 0 c 2 1 2 = 0 v d dt p d dt m 0 c 2 1 2 = 0
d dt p = F d dt m 0 1 2 c 2 = vf = d dt E
E = mc 2 m = m 0 1 2 m 0 m 0 c 2
E = mc 2 = 2m 0 c 2 = m 0 c 2 m 0 1 2 c 2 1 2 1 2 = 0.25 2 = 0.75 = v c = 3 2
E k = E m 0 c 2 = m 0 c 2 ( 1 1) 1 2
1 1 x =1+ 1 2 x + 3 8 x 2 + x << 1 1 1 x =1+ 1 2 x
E k = E m 0 c 2 = m 0 c 2 ( 1 1) 1 2 = m 0 c 2 ( 1 2 v 2 c 2 + 3 8 v 4 c 4 +) = 1 2 m 0v 2 + 3 8 m 0 v 4 c 2 +
if v c << 1 E k = m 0 c 2 ( 1 1) 1 2 = 1 2 m 0v 2
E = mc 2 E 2 = m 2 c 4 0 1 2 = m 2 c 4 (1 2 ) + m 2 c 2 v 2 0 0 1 2 = m 0 2 c 4 + c 2 m 0 2 1 2 v 2 E = m 0 2 c 4 + c 2 p 2
I( f,t) = 2hf 3 h = 6.626176 10 34 [J Hz -1 ] c 2 1 e hf /kt 1
E = hf n (n = 1,2,3,4,) h = 6.626176 10 34 [J Hz -1 ]
0.9109534 10 30 kg e e = 1.6021892 10 19 C 1.6726485 10 27 kg e
h = 6.626176 10 34 [J Hz -1 ] hf
5.64 10 14 Hz 5.64 10 14 Hz
E = hf n (n = 1,2,3,4,) h = 6.626176 10 34 [J Hz -1 ]
E = hf h = 6.626176 10 34 [J Hz -1 ]
n X [W] f [Hz] X = nhf h = 6.626176 10 34 [J Hz -1 ]
n (1 / n) [s]
(1 / n) [s]
h = 6.626176 10 34 [J Hz -1 ] hf
hf h = 6.626176 10 34 [J Hz -1 ]
c = 2.99792458 10 8 [m / s] h = 6.626176 10 34 [J Hz -1 ] f = c = 2.998 108 532 10 9 = 5.64 10 14 Hz E = hf = 3.74 10-19 J
E = hf = 3.74 10-19 J 1mW =110-3 [J / s] N[1/ s] = 110-3 [1/ s] 3.74 10-19 = 2.67 10 15