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ddx
x 2 = 2x
d
dxx 3 = 3x 2
ddx
xn = nxn 1
xa
b dx =
12b2
12a2
x 2a
b dx =
13b3
13a3
xna
b dx =
1n +1
bn+1 1n +1
an+1
ddx
f (ax) = a f (ax)
d
dxf (x) =
0lim
f (x + ) f (x)
ddx
f (ax) =0
limf (a(x + )) f (ax)
=0
limf (ax + a ) f (ax)
=0
a limf (ax + a ) f (ax)
a
=(a ) 0
a limf (ax + a ) f (ax)
a
= a f (ax)
d
dx(2x)3 = 2 3 (2x)2 = 24x 2
ddx(1+ 2x)3 = 2 3 (1+ 2x)2 = 6(1+2x)2
360o = 2 rad
180o = rad
60o =3
rad
sin(x)
cos(x)
sin2(x) + cos2(x) =1
sin(6
) =12
cos(4
) =1
2
sin(43
) = sin( +13
)
= sin(1
3)
=3
2
y1 = ax1 + bx2y2 = cx1 + dx2
y1y2
=
a b
c d
x1x2
A =a b
c d
y1y2
= A
x1x2
A =a b
c d
y1y2
= A
x1x2
x1x2
= B
y1y2
B =u11 u12u21 u22
=
1ad bc
d b
c a
u11 =d
ad bc u22 =
a
ad bc
u12 =b
ad bc u21 =
c
ad bc
A =a b
c d
y1y2
= A
x1x2
y1 = ax1 + bx2y2 = cx1 + dx2
dy1 = adx1 + bdx2by2 = bcx1 + bdx2
dy1 by2 = adx1 bcx1
x1 =d
ad bcy1 +
b
ad bcy2
cy1 = acx1 + bcx2ay2 = acx1 + adx2
ay2 cy1 = adx2 bcx2
x2 =c
ad bcy1 +
a
ad bcy2
34000 = 5X + 3Y
42000 = 6X + 4Y
34000
42000
=
5 3
6 4
X
Y
X
Y
=
42
32
6
2
5
2
34000
42000
=2 -1.5
-3 2.5
34000
42000
=2 34000 1.5 42000
3 34000 + 2.5 42000
=5000
3000
A =a11 a12a21 a22
B =b11 b12b21 b22
y1y2
= A
x1x2
z1z2
= B
y1y2
z1z2
= BA
x1x2
BA =b11a11 + b12a21 b11a12 + b12a22b21a11 + b22a21 b21a12 + b22a22
y1 = a11x1 + a12x2y2 = a21x1 + a22x2z1 = b11y1 + b12y2z2 = b21y1 + b22y2
z1 = b11 (a11x1 + a12x2)+ b12(a21x1 + a22x2)
z2 = b21 (a11x1 + a12x2)+ b22(a21x1 + a22x2)
z1 = (b11a11 + b12a21 )x1 + (b11a12 + b12a22)x2z2 = (b21a11 + b22a21 )x1 + (b21a12 + b22a22)x2
z1z2
=
b11a11 + b12a21 b11a12 + b12a22b21a11 + b22a21 b21a12 + b22a22
x1x2
X1X2Y1Y2Z1Z2
Y1Y2
=
5 3
6 4
X1X2
Z1Z2
=
2 4
5 4
Y1Y2
Z1Z2
=2 4
5 4
5 3
6 4
X1X2
=2 5 + 4 6 2 3 + 4 4
5 5 + 4 6 5 3 + 4 4
X1X2
=34 22
49 31
X1X2
V1
V0
V2
f00V0 = f0 0
(V0 V1 ) T f0 T
=V0 V1f0
f '=V0
=
V0V0 V1
f0
T (V0 + V2) T
0
=V0 T
0
V0 + V2V0
f =V0 + V2
V0f0
f =V0 + V2V0 V1
f0
V0 = 340m / s
V1 = ±19.0827m / s = ±68.70km /h
V2 = 0
f± =340
340 m19.0827f0
f+ =1.0595 f0f = 0.9469 f0
21
12 =1.0595
v =1 [m / s]
t [s]
x [m]
012345
0 1 2 3 4 5
t [s]
x [m]
012345
0 1 2 3 4 5
0s 2s v =4 - 02 - 0
[m / s] = 2 [m / s]
2s 3s v =2 - 43 - 2
[m / s] = -2 [m / s]
3s 5s v =5 - 25 - 3
[m / s] =1.5 [m / s]
t [s]
x [m]
012345
0 1 2 3 4 5
0s 1s v =2 - 01- 0
[m / s] = 2 [m / s]
1s 2s v =4 - 22 -1
[m / s] = 2 [m / s]
2s 3s v =2 - 43 - 2
[m / s] = -2 [m / s]
3s 4s v =3.5 - 24 - 3
[m / s] =1.5 [m / s]
4s 5s v =5 - 3.55 - 4
[m / s] =1.5 [m / s]
t [s]
x [m]
012345
0 1 2 3 4 5
1s 2s
v =3 1
2 1 [m / s] = 3 [m / s]
t[s] (t+ )[s]
v =x(t + ) x(t)
(t + ) t[m / s] =
x(t + ) x(t)[m / s]
t
v(t) =0
limx(t + ) x(x)
=dx
dt
v(t) =dxdt
x(t) = a0 + a1t + a2t2
+ a3t3
v(t) =ddtx(t) = a1 + 2a2t + 3a3t
2
d
dxF(x) = f (x) F(x) f(x)
f (x)a
b dx = F(b) F(a)
d
dtx(t) = v(t)
v(t)t1
t2 dt = x(t2) x(t1 )
t=0 x=0
t
v(t) = b0 + b1t + b2t2
t
x(t) = x(0) + (b0 + b1 t + b2t2)
0
t dt
= b0t +1
2b1 t
2+
1
3b2t
3
d
dtv(t)
a(t) =d
dtv(t)
v(t) =d
dtx(t)
a(t) =ddt
ddtx(t)
=d2
dt 2x(t)
v(t) = c0 + c1t + c2t2
+ c3t3
a(t) =ddtv(t) = c1 + 2c2t + 3c3t
2
x(t) = b0 + b1t + b2t2
+ b3t3
+ b4t4
v(t) = b1 + 2b2t + 3b3t2
+ 4b4t3
a(t) = 2b2 + 6b3t +12b4t2
a(t) =d
dtv(t)
a(t) dtt1
t2= v(t2) v(t1 )
v(t) = v(t1 ) + a(t) dtt1
t
t=0 x=0
t=0
a(t) = b0 + b1 t + b2t2
v(t) = v0 + (b0 + b1t + b2t2)
0
t dt
= v0 + b0t +12b1 t
2 +13b2t
3
x(t) = v0t +12b0t
2 +12
13b1t
3 +13
14b2t
4
= v0t +12b0t
2+
16b1 t
3+
112b2t
4
m
Fm
Fm1m1
1
2Fm1
m1 =1
2m
mv
p = mv
p = mv
ddtp = F
F =ddtp
F(t) t1
t2dt = p(t2) p(t1 )
v(t2) =m(t1 )
m(t2)v(t1 ) +
1m(t2)
F(t)t1
t2 dt
p = mv
ddtp = F
md
dtv(t) = F
ma(t) = F
a(t) =1
mF
t=0 x=0
t=0
F(t) = c0 + c1 t
v(t) =1
m(c0t +
1
2c1t
2)
x(t) =1m(12c0t
2+16c1t
3)
m T
u(t,n)u(t,n + 1)
u(t,n 1)
n
md2
dt 2u(t,n) = T sin( n ) T sin( n 1 )
C
xy
sin( )
=y
x 2 + y 2
=1
1+y
x
2 y
x
y
x
y
x
<< 1
md2
dt 2u(t,n)
= T sin( n ) T sin( n 1 )
= Tu(t,n + 1) u(t,n)
CTu(t,n) u(t,n 1)
C
= Tu(t,n + 1) u(t,n)
CTu(t,n) u(t,n 1)
C
= Tu(t,x +C) u(t,x)
C
u(t,x) u(t,x C)C
(x = nC)
tu(t,x) =
0Lim
u(t + ,x) u(t,x)
xu(t,x) =
0Lim
u(t,x + ) u(t,x)
f (x,y) = c1x + c2x2
+ c3xy3
xf (x,y) = c1 + 2c2x + c3y
3
yf (x,y) = 3c3xy
2
C 0
m2
t 2u(t,x)
= TC 0Lim
u(t,x + C) u(t,x)C
u(t,x) u(t,x C)C
= Txu(t,x)
xu(t,x C)
= TCC 0Lim
1C x
u(t,x)xu(t,x C)
= TC2
x 2u(t,x)
m2
t 2u(t,x) = TC
2
x 2u(t,x)
2
t 2u(t,x) = T
1 2
x 2u(t,x)
2
t 2u(t,x) =
T 2
x 2u(t,x)
=m
C
2
t 2u(t,x) = v 2
2
x 2u(t,x)
ddxsin(x) = cos(x)
ddxcos(x) = sin(x)
u(t,x) = sin(k(x vt))
tu(t,x) = kv cos(k(x vt))
2
t 2u(t,x) = ( kv)2 sin(k(x vt))
xu(t,x) = k cos(k(x vt))
2
x 2u(t,x) = (k)2 sin(k(x vt))
2
t 2u(t,x) = v 2
2
x 2u(t,x)
sin(k(x vt))
t = 0
t 0
x
xvt
sin(k(x vt)) v
f
sin(k(x vt))
= sin(kx vkt)
= sin(kx t)
k =2
= 2 f
2 f =
= vk
v = f
sin(k(x vt))
= sin(kx vkt)
= sin(kx t)
k =2
= 2 f
2
t 2u(t,x) =
T 2
x 2u(t,x)
=m
C
v = ±T
sin(k(x + vt))
sin(k(x vt)) vv
2
t 2u1 (t,x) = v 2
2
x 2u1 (t,x)
2
t 2u2(t,x) = v 2
2
x 2u2(t,x)
u3(t,x) = a1u1 (t,x) + a2u2(t,x)
2
t 2u3(t,x) = a1
2
t 2u1 (t, x) + a2
2
t 2u2(t,x)
= a1v2
2
x 2u1 (t,x)+ a2v
22
x 2u2(t,x)
= v 2(a12
x 2u1 (t,x)+ a2
2
x 2u2(t,x))
2
x 2u3(t,x) = a1
2
x 2u1 (t,x) + a2
2
x 2u2(t,x)
2
t 2u1 (t,x) = v 2
2
x 2u1 (t,x)
2
t 2u2(t,x) = v 2
2
x 2u2(t,x)
u3(t,x) = a1u1 (t,x)+ a2u2(t,x)
2
t 2u3(t,x) = v 2
2
x 2u3(t,x)
L
x =L
2x =
L
2
u(t,L
2) = 0
u(t,L
2) = 0
u(t,x) = a1 sin(kx t) + a2 sin(kx + t)
u(t,L
2)
= a1 sin(L
2k t) + a2 sin(
L
2k + t) = 0
u(t,L
2)
= a1 sin(L
2k t) + a2 sin(
L
2k + t) = 0
sin(x1 + x2) = sin x1 cos x2 + cos x1 sin x2
cos(x1 + x2) = sin x1 sin x2 cos x1 cos x2
0 = a1 sin(L
2k t) + a2 sin(
L
2k + t)
= a1 sin(L
2k)cos t a1 cos(
L
2k)sin t
a2 sin(L
2k)cos t + a2 cos(
L
2k)sin t
0 = a1 sin(L
2k t) + a2 sin(
L
2k + t)
= a1 sin(L
2k)cos t a1 cos(
L
2k)sin t
+ a2 sin(L
2k)cos t + a2 cos(
L
2k)sin t
A + B2
(a2 a1 )cos(Lk
2)sin t = 0
B- A2
(a2 + a1 )sin(Lk
2)cos t = 0
(a2 a1 )cos(Lk
2) = 0
(a2 + a1 )sin(Lk
2) = 0
(a2 a1 ) 0 cos(Lk
2) = 0 sin(
Lk
2) 0 (a2 + a1 ) = 0
cos(Lk
2) 0 (a2 a1 ) = 0 (a2 + a1 ) 0 sin(
Lk
2) = 0
cos(Lk
2) = 0 and (a2 + a1 ) = 0
OR
sin(Lk
2) = 0 and (a2 a1 ) = 0
(a2 + a1 ) = 0
u(t,x) = a1 (sin(kx t) sin(kx + t))
= 2a1 coskx sin t
cos(kL
2) = 0
kL
2=12
, 32
, 52
k =L
, 3L
, 5L
, 7L
(a2 a1 ) = 0
u(t,x) = a1 (sin(kx t) + sin(kx + t))
= 2a1 sinkxcos t
sin(kL
2) = 0
kL
2= , 2 , 3
k = 2L
, 4L
, 6L
, 8L
= vk
f =2
=12
T k =
12L
Tn
(n = 1, 2, 3, 4, L)
L =12 f
T
f1 =12L
T1
f2 =12L
T2
f2f1
=T2T1
f2f1
= 1.1
T2T1
= 1.21
•r E = 0
•r B = 0r E =
t
r B
r B = μ0 0 t
r E
r E r B
0
μ0
r E Q
r F = Q
r E
r A = (Ax, Ay, Az )
•r A =
xAx +
yAy +
zAz
r A = (
yAz z
Ay, zAx x
Az, xAy y
Ax )
U = (x
U, y
U, z
U)
0 =107
4 c 2 [C2 / Nm2] = 8.85418782 10-12 [C2 / Nm2]
μ0 = 4 10 7 [N / A2] =1.25663706 10-6 [N / A2]
c = 2.99792458 108[m / s]
(r A ) = ( •
r A ) 2
r A
2r A = (
2
x 2Ax +
2
y 2Ax +
2
z2Ax,
2
x 2Ay +
2
y 2Ay +
2
z2Ay,
2
x 2Az +
2
y 2Az +
2
z2Az )
(r E ) =
t
r B
r B = μ0 0 t
r E
•r E = 0
•r B = 0r E =
t
r B
r B = μ0 0 t
r E
(r E ) = ( •
r E ) 2
r E = μ0 0
2
t 2r E
2r E = μ0 0
2
t 2r E
2r E = μ0 0
2
t 2r E
r E = (0,Ey (x),0)
yEy = 0
zEz = 0
2
x 2Ey = μ0 0
2
t 2Ey
2
t 2Ey =
1μ0 0
2
x 2Ey
2
t 2Ey = c 2
2
x 2Ey
D
X
DD
X
X = n
n = 1,2,3,4LD
X
X = Dsin
sin =Dn
= 532 nm = 5.32 10-7m= 650 nm = 6.50 10-7m
D = 0.02 mm = 2.00 10 5m
= 5.32 10-7m,D
= 0.0266
n =1 sin = 0.0266 = 0.026603 rad =1.5242o
n = 2 sin = 0.0532 = 0.053225 rad = 3.0496o
= 6.50 10-7m,D
= 0.0325
n =1 sin = 0.0325 = 0.032506 rad =1.8624 o
n = 2 sin = 0.0650 = 0.065046 rad = 3.7269o
V = 3.0 104 [m / s] =1.1 105[km / h]
L
L
(VT1 )2
+ L2 = (cT1 )2
T1 =2L
c 2 V 2
T2 =L
c +V+
L
c V=2cL
c 2 V 2
T2 T1
=2cL
c 2 V 2
2L
c 2 V 2
=2L
c 2 V 2(
1
1 (V
c)2
1)
=2L
c 1 (V
c)2
1
1 (V
c)2
1
V = 0
c = 2.99792458 108[m / s]
x
x
t
t v
x
x
t
t
t = t
t = 0
tt = t
t 0
x
t
vt
x = x vt
x
v
x = x vt
x
t
=1 v
0 1
x
t
t=0 x m
x m
x = 9.8 + 0.1t x = x 5t x = 0
0 = 9.8 + 0.1t 5t
4.9t = 9.8
t = 2
x
t
=
1 v10 1
x
t
x
t
=
1 v20 1
x
t
x
t
=
1 v20 1
1 v10 1
x
t
=1 1 v2 0 1 (-v1) v2 1
0 1 +1 0 0 (-v1) +1 1
x
t
=1 (v1 + v2)
0 1
x
t
v1 + c c
c + v2 c
x 2 + y 2 + z2 = (ct)2
( x )2 + ( y )2 + ( z )2 = (c t )2
y = y
z = z
x
t
=
u11 u12u21 u22
x
t
y = y
z = z
x
t
=
u11 u12u21 u22
x
t
x
x
x = 0d
dtx = v
0 = u11x + u12t
x =u12u11
t
v
u12u11
= v
x
t
=
u11 u12u21 u22
x
t
x = 0d
d t x = v
x = u12t
t = u22t
x =u12u22
t
x
x
v
u12u22
= v
u22 = u11
x 2 + y 2 + z2 = (ct)2
( x )2 + ( y )2 + ( z )2 = (c t )2
x
t
=
u11 vu11u21 u11
x
t
( x )2 + ( y )2 + ( z )2 = (c t )2
(u11x vu11t)2
+ y 2 + z2 = c 2(u21x + u11t)2
u112 x 2 2vu11
2 xt + v 2u112 t 2 + y 2 + z2
= c 2u212 x 2 + 2c 2u21u11xt + c 2u11
2 t 2
(u112 c 2u21
2 )x 2 + y 2 + z2 2(vu112
+ c 2u21u11 )xt
= (c 2u112 v 2u11
2 )t 2
(u112 c 2u21
2 ) = 1 (1)
(c 2u112 v 2u11
2 ) = c 2 (2)
(vu112
+ c 2u21u11 ) = 0 (3)
(2) u11 =c
c 2 v 2=
1
1 (v
c)2
(3) u21 =v
c 2u11 =
(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
x
x
v
x
w
=
x
w
w = ct
w = c t
=v
c
=1
1 2
1 =v1c
, 2 =v2c
1 =1
1 12
, 2 =1
1 22
x
w
=
1 1 1
1 1 1
x
w
x
w
=
2 2 2
2 2 2
x
w
x
w
=
2 2 2
2 2 2
1 1 1
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 + 1222
12
22 2 1 2
=(1+ 1 2)
1+ 1222
12
22
=(1+ 1 2)
(1 12)(1 2
2)
= 1 2(1+ 1 2)
x
w
=
1 2(1+ 1 2) ( 1 + 2) 1 2
( 1 + 2) 1 2 1 2(1+ 1 2)
x
w
=
x
w
= 1 + 2
1+ 1 2
=1
1 2 =v
cv = c
v =v1 + v2
1+v1v2c 2
v2 = c
v =v1 + c
1+v1c
c 2
=v1 + c
1+v1c
= c
v = c
v
v =v1 + v2
1+v1 v2c 2
=0.7c + 0.5c1+ 0.35
=1.2c1.35
=2427c =
89c
x
w
=
x
w
w = ct
w = c t
=v
c
=1
1 2
x
x
v
x
w
=
x
w
w = ct
w = c t
=v
c
=1
1 2
x
x
v
x
w
=
x
w
w = ct
w = c t
=v
c
=1
1 2
L0 ( x , t ) L0
x 2 x 1 = L0
x = x 2 x = x 1
( x , t )
x v
L0
x2 x1 = L
x = x1
(x,t)
x
x = x2
t = t0
x = x1 x = x2
t = t0
x
w
=
x
w
t 1 =c
x1 + t0
t 2 =c
x2 + t0
x2 x1 t 2 t 1
x = x1 x = x2
t = t0
x
w
=
x
w
x 1 = x1 c t0 x 2 = x2 c t0 x 2 x 1 = (x2 x1 )
L0 = L
L = 1 2 L0
L < L0
x
=v
c= 0.8
1 2= 1 0.64 = 0.6
L = 0.6 5 = 3.0
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 = 0.5cv =
0.5c 0.5c1+ 0.5 0.5
=c
1+ 0.25= 0.8c
1 2= 1 0.82 = 0.6
L = 0.6103
= 2 3 m (= 3.46410 m)
1 0.5 0.5 L0 = 5
L0 =50.75
=103
m
L = 1 0.5 0.5 5 = 5 0.75 = 2.5 3 m
x
x
v
t
t
x = x 0 t
A =a b
c d
y1y2
= A
x1x2
x1x2
= B
y1y2
B =u11 u12u21 u22
=
1ad bc
d b
c a
u11 =d
ad bc u22 =
a
ad bc
u12 =b
ad bc u21 =
c
ad bc
x
w
=
x
w
w = ct
w = c t
=v
c
=1
1 2
=2(1 2) =1
x
w
=
x
w
x
x
v
t
t
x = x 0 t
x = x 0 + c t
ct = x 0 + c t
x = x 0 + c t
ct = x 0 + c t ct1 = x 0 + c t 1ct2 = x 0 + c t 2t2 t1 = ( t 2 t 1 )
t = t w = ct
w = c t
=v
c
=1
1 2
t = t2 t1 t = t 2 t 1
t =1
1 2 t
t > t
x = vt
t =1
1v
c
2 t
t =1
1 0.64 t =
1
0.6 t =
5
3 t
t = 500 s
(X1 , Y1, Z1 ), (X2, Y2, Z2), (X3, Y3, Z3), (X4, Y4, Z4)
T1, T2, T3, T4
(x,y,z)t t
[c(t + t T1 )]2
= (X1 x)2 + (Y1 y)2 + (Z1 z)2
[c(t + t T2)]2
= (X2 x)2 + (Y2 y)2 + (Z2 z)2
[c(t + t T3)]2
= (X3 x)2 + (Y3 y)2 + (Z3 z)2
[c(t + t T4)]2
= (X4 x)2 + (Y4 y)2 + (Z4 z)2
T1, T2, T3, T4
t T1, t T2, t T3, t T4
x L
T =2L
c
xL
T
Tv
˜ L = 2 L2 +Tv
2
2
T =˜ L
c
˜ L = 2 L2 +Tv
2
2
T =˜ L
c
cT = 2 L2 +Tv
2
2
c 2T 2= 4(L2 +
Tv
2
2
)
(c2 v 2)T 2= 4L2
T =2L
c2 v 2=
2L
c 1v
c
2
T =2L
c
T =1
1v
c
2 T
T =1
1 2 T
c = 2.99792458 108[m / s]
sin(kx t) = sin(2
x 2 f t)
= sin(2cf (x ct))
sin(2
cf (x + ct))
v v
sin(2
cf (x + ct))sin(
2
cf (x ct))
sin(2
cf (x ct))sin(
2
cf0( x c t ))
x
c t
=
x
ct
sin(
2c
f0( x c t ))
= sin(2c
f0(( x ct) ( x + ct)))
= sin(2c
f0(( + )x ( + )ct))
= sin(2
cf0 (1+ )(x ct)) f = (1+ ) f0
f = (1+ ) f0
=1+
1 2f0
=(1+ )2
(1 )(1+ ) f0
=1+
1 f0
sin(2
cf (x + ct))sin(
2
cf0( x + c t ))
x
c t
=
x
ct
sin(2c
f0( x + c t ))
= sin(2c
f0(( x ct)+ ( x + ct)))
= sin(2c
f0(( )x ( )ct))
= sin(2
cf0 (1 )(x + ct))
f = (1 ) f0
f = (1 ) f0
=1
1 2f0
=(1 )2
(1 )(1+ ) f0
=11+
f0
f =1+
1 f0
> 0< 0
f =V0 +V2V0 V1
f0
=1+ (V2 /V0)1 (V1 /V0)
f0
10 THz = 1013 Hz
= 0.5
f =1+
1f0
=1.50.5
f0
= 3 f0 = 3 1013Hz
v
f0
f2f1
f1 =1+
1 f0
f2 =1+
1 f1 =
1+
1 f0
if |R |< 1
S =11 R
= 1+ R + R2 + R3 +
f2 =1+
1 f0
= (1+ )(1+ +2
+3
+ ) f0= (1+ 2 + 2 2
+ 2 3+ ) f0
f2 f0 = 2( +2
+3
+ ) f0
if << 1
f2 f0 = 2 f0
v =100 km / h = 27.7778 m / s
f0 =10GHz =1010Hzc = 2.99792458 108 m / s
=v
c= 9.2656693 10-8
f2 f0 = f = 2 f0 =1853.13Hz
=1.853KHz
x
y
x
y
D y =sin
cos x
( y y0)sin + ( x x0)cos = 0
y0 = D sin
x0 = D cos ( y D sin )sin + ( x D cos )cos = 0
y sin + x cos = D sin2 + D cos2
D = y sin + x cos
y
x
sin(2
cf0( D + c t ))
= sin(2c
f0( x cos + y sin + c t ))
sin(2
cf0( x cos + y sin + c t )) y
x
y
x
sin(2
cf (x cos + y sin + ct))
x = x ct
c t = x + ct
sin(2c
f0( x cos + y sin + c t ))
= sin(2
cf0(( x ct)cos + y sin + ( x + ct)))
= sin(2c
f0(( cos )x + y sin + ( cos + )ct))
= sin(2
cf0( (cos )x + y sin + ( cos + 1)ct))
= sin(2c
f0 (1 cos )(cos
1 cos x +
sin
(1 cos )y + ct))
sin(2
cf0( x cos + y sin + c t ))
= sin(2c
f0 (1 cos )(cos
1 cos x +
sin
(1 cos )y + ct))
sin(2c
f (x cos + y sin + ct))
f = (1 cos ) f0
cos =cos
1 cos
sin =sin
(1 cos )
cos2 + sin2 =cos
1 cos
2
+sin
(1 cos )
2
=(cos )2 + (1 2)sin2
(1 cos )2
=cos2 2 cos +
2+ sin2 2 sin2
(1 cos )2
=1 2 cos +
2 2 sin2
(1 cos )2= 1
cos =cos
1 cos sin =
sin
(1 cos )
=0=0.2=0.5=0.8=0.9
=0.99
cos =cos
1 cos
if =2
= 90
cos = 0
cos = =v
c
=2
sin =v
c
v
V0
tan =v
V0
f = (1 cos ) f0
cos =cos
1 cos
if =2
cos = 0
cos = 0
f = (1 2) f0 = 1 2 f0
y
x
f = 1 2 f0
ddtP = F
dt2t0
t ddt2
P = dt2t0
t F
p(t) p(t0) = dt2t0
t F
p = m0v
Fp(t) p(t0) = dt2t0
t F = (t t0)F
m0
p = m0v
v(t) v(t0) = v =t t0m0
F
v(t) = v(t0) + v
v =v1 + v2
1+v1v2c 2
v =tF
m0
v(t) < v(t0) + v
p(t) p(t0) = p = t F
x x
v ( t 0) = 0
t = t t 0 v = v ( t ) v ( t 0)
p = m0 v = t F
p = m0 v = t F
v(t0) = v
p = tF
t = t
1 2
p = t
1 2F =
m0 v
1 2
p = (m + m)(v + v) mv = v m + m v
v =v + v
1+v v
c 2
v
= (v + v )(1v v
c 2
+
v v
c 2
2
) v
= v v 2
c 2
v + v
v v
c 2
2
v v v
c 2
= (1 2) v
v =1
1 2 v
v m + m v = m0(12) (3/2) v
m =v
vm0(1
2) (3/2)(1m
m0
(1 2)(3/2))
m
v=1vm0(1
2) (3/2)(1m
m0
(1 2)(3/2))
0
dmdv
=1vm0(1
2) (3/2)(1m
m0
(1 2)(3/2))
dmdv
=1vm0(1
2) (3/2)(1m
m0
(1 2)(3/2))
if m =m0
1 2=
m0
1 (v / c)2
ddvm =
12m0(1 (v / c)2) (3/2)( 2
v
c 2)
=1vm0(1 (v / c)2) (3/2) 2
(1m
m0
(1 2)(3/2))
= 1 (1 2) =2
m =m0
1 2
=v
c
p = mv =m0
1 2v
m =m0
1 2
m0
m =m0
1 2=
601 0.64
=600.6
= 100kg
F
x
E = Fx
ddtE = F
ddtx = Fv
u1 =v
1 2
u2 =c
1 2
u12 u2
2=v 2 c 2
1 2 =
c 2(v
c
2
1)
1v
c
2
= c 2
u12 u2
2= c 2
d
dt(u1
2 u22) = 0
2u1 (ddtu1 ) 2u2(
ddtu2) = 0
2u1 (ddtu1 ) 2u2(
ddtu2) = 0
v
1 2
ddt
m0
1 2v
1
1 2
ddt
m0
1 2c 2
= 0
vddtp
ddt
m0
1 2c 2
= 0
ddtp = F
d
dt
m0
1 2c 2 = vF =
d
dtE
E = mc 2
m =m0
1 2
m0
m0c2
1 2= 0.25
2= 0.75
=v
c=32
E = mc 2 =m0
1 2c 2
2m0c2
=m0c
2
1 2
Ek = E m0c2
= m0c2(
1
1 21)
11 x
=1+12x +
38x 2 +
| x |<< 1
11 x
=1+12x
Ek = E m0c2
= m0c2(
1
1 21)
= m0c2(12v 2
c 2+38v 4
c 4+ )
=12m0v
2+3
8m0
v 4
c 2+
if |v
c|<< 1
Ek = m0c2(
1
1 21)
=12m0v
2
E = mc 2
E 2=m02c 4
1 2 =m02c 4(1 2)+ m0
2c 2v 2
1 2
= m02c 4 + c 2
m02
1 2 v2
E = m02c 4 + c 2p2
I( f ,T) =2hf 3
c 21
ehf /kT 1h = 6.626176 10 34[J •Hz-1]
E = hf n
(n = 1,2,3,4, )
h = 6.626176 10 34[J •Hz-1]
0.9109534 10 30kge
e = 1.6021892 10 19C
1.6726485 10 27kg
e
h f
h = 6.626176 10 34[J •Hz-1]
5.64 1014 Hz
5.64 1014 Hz
E = hf n (n = 1,2,3,4, )h = 6.626176 10 34[J •Hz-1]
E = hfh = 6.626176 10 34[J •Hz-1]
nX [W]
X = nhf
f [Hz]
h = 6.626176 10 34[J •Hz-1]
n
(1 / n) [s]
(1 / n) [s]
h f
h = 6.626176 10 34[J •Hz-1]
hf
h = 6.626176 10 34[J •Hz-1]
c = 2.99792458 108[m / s]
h = 6.626176 10 34[J •Hz-1]
f =c
=2.998 108
532 10 9 = 5.64 1014Hz
E = hf = 3.74 10-19 J
E = hf = 3.74 10-19 J
1mW =1 10-3 [J / s]
N[1 / s] =1 10-3
3.74 10-19 [1 / s]
= 2.67 1015
