( )
ńł
�ł1 i = j
�ł
�ij a"
�ł
ół0 i = j

"
a = aięi = aięi
i=1
ńł
�ł
�ł1
�ł
�ł
�ł
�ł
�ijk
�ł-1
�ł
�ł
�ł
�ł
ół0
a � b = aibi, ęięj = �ij
a � b = �ijkęiajbk
a � b � c = �ijkęiaj�klmękblcm = ęibi ajcj - ęici ajbj
c
b (a�c)
a� b
( )
a � b � c = (a � c) � b - a � b � c
W ektory : a = (a1, a2, a3) , b = (b1, b2, b3)
a + b = (a1 + b1, a2 + b2, a3 + b3)
a - b = (a1 - b1, a2 - b2, a3 - b3)
a � b = |a| | b|cosą = a1b1 + a2b2 + a3b3
b
a � b = - b � a
x w
ę
Ć
AAAA a � b = Ć
a1 a2 a3 = x (a2b3 - a3b2) + w
(a3b1 - a1b3) + ę (a1b2 - a2b1)
b1 b2 b3
AAAA |a � b| = |a| � | b| � sin ą =
a1 a2 a3
AAAA a � b � c =
b1 b2 b3 =
a, b, c
c1 c2 c3
r
dr
v a" a" Y
dt
"r
v =
"t
dv d2r
Ł �
a = = v = = r
dt dt2
"v
a =
"t
r = (x, y) - (x, w
)
Ć
r = xx + yw

Ć
ńł
�łv = �
�ł
x
dr d
v = = (xx + yw
) = �x + Źw
= (�, Ź)
Ć Ć
dt dt �ł
ółvy = Ź
ńł
�ła = ć
�ł
x
dv d
a = = (�x + Źw
) = (ć, �)
Ć
�ł
dt dt
ółay = �
r = rnr + 0n� = (r, 0)
Ć Ć
ńł ńł
�łx = r � cos� �łr = x2 + y2
�ł �ł
�ł
óły = r � sin � �ł = y
ółtg�
x
d
v = Y = (r � �r) = Y�r + r�r
Ł
dt
�r =?
Ł
"
"
"ur d�
�r = lim = �� = ���
Ł Ł
"t dt
"ur = | �r | "�
=1
v = Y�r + r��� = (Y, r�)
Ł Ł
vr = Y, v� = r� ( vl = r�)
Ł
2 2
v = vr + v� = Y2 + r2�2
Ł
d
Ł
a = v = (Y�r + r���) = r�r + Y�r + Y��� + r��� + r���
Ł � Ł Ł � Ł Ł
dt
=1
�r = ��� "u� = | �� | "�
Ł Ł
"u�
�� = lim = -��r
Ł Ł
"t
a = r�r + Y��� + Y��� + r��� - r�2�r =
� Ł Ł � Ł
= (r - r�2)�r + (r� + 2Y�)��
� Ł � Ł
ńł
�ła = r - r�2 ( )
�ł
� Ł
r
�ł
óła� = r� + 2Y� ( )
� Ł
a = a2 + a2
r �
" ad = r�2
ad = � � (� � r ) �
" ac = 2wv
ac = 2� � v acĄ"v
v = v�t + 0 � �n = v�t
dv
a = = v�t + v �t = v�t + annu
Ł Ł Ł Ć
dt
�n
at = v
Ł
v2 v2
an = �! � =
� au
v = at, � = ct
 v = const, a = 0, s = v � t
a = const,
dv
v t
= a �! dV = a � dt �! v - v0 = at �! v = v0 + at
v0 0
dt
S
v0 + (v0 + at) at2
S = t = v0t +
2 2
ds
s t
= v = v0 + at ds = (v0 + at)dt
0 0
dt
gt2 2h
h = �! t =
2 g
"
vk = gt = 2gh
v0
v = v0 - gtw �! tw =
g
2
gt2 v0
w
h = v0tw - =
2 2g
gt2 2h
h = �! t =
2 g
2h
d = v0t = v0
g
2 2 2
v2 = v0 + vy = v0 + (gt)2
2
v2 = v0 + 2gh
vy
tgą =
v0
vy = v0y - gtw = 0
voy v0 sin ą
tw = =
g g
tc = 2tw
2
2v0 sin ą cos ą
d = v0xtc =
g
2
v0 sin(2ą)
d =
g
d(ą) = d(90ć% - ą)
2 2
gt2 gv0 sin2 ą v0 sin2 ą
w
h = = =
2 2g2 2g
ńł ńł
�łx = v0xt �łdx = v0x
�ł �ł
x
dt
t =
�ł vox ół
�łdy
óły = v0yt - gt2
= v0y - gt
2 dt
x g x2
y = v0y - �!
2
v0x 2 v0x
dvy v0y gx d
= 0 - = 0 x =
2
dx vox vox 2
2
d 2v0xvoĆ v0 sin 2ą
v0xv0y = g �! d = =
2 g g
dr
s = rĆ v =
dt
ds d(rĆ) dĆ
v = = = r = r�
dt dt dt
v = � � r
dv d d� dr
a = = (� � r) = � r + � �
dt dt dt dt
v
d�
a = �r + � � v = � � r + � � v
dt
ad
�
ad = � � v = � � (� � r) = (� � r) �� - (� � �) � r
=0 �Ą"r
v2
ad = -�2r = - r
Ć
r
2Ą
� = = 2Ąf
T
�
T
f
�t2
� = �0t + � = �0t + �t
2
v2
as = �r, ad = �2r =
r
 �
dr dr
v = = r = x ę1 + y ę2 + z ę3
dt dt
dx dy dz
v = ę1 + ę2 + ę3
dt dt dt
dę1
= x = � � ę1 (v = � � r)
dt
�ł �ł
dr
�łx ę1 + y ę2 + z ę3 łł
= v + � �
dt
=r
v = v + � � r
v = 0 v = ��r (r = cost)
d d
= + ��
dt dt
v = v + � � r
dv
a =
dt
dv d
a = = (v + � � r ) + � (v + � � r ) =
dt dt
dv dr
= + � � + � � v + � � (� � r ) =
dt dt
(��r)��-�2r
= a + 2� � v + � � (� � r )
�Ą"r Ą" �
a = a + 2� � v - �2r
ac = 2� � v
ad = � � (� � r )
 F = ma
dv m � dv dp
a = F = =
dt dt dt
dp
F =
dt
F dt = dp
dm
� =
dt
F
dp d dm dv
F = = (mv) = v +m
dt dt dt dt
=�
m = m0 + �t
dv
F = (m0 + �t) + �v
dt
dv
F - �v = (m0 + �t) +
dt
dv dt dv dt
v t
= =
0 0
F - �v m0 + �t F - �v m0 + �t
1 1
- ln(F - �v)|v = ln(m0 + �t)|t
0 0
� �
F - �v m0 + �t
-ln = ln
F m0
F m0 + �t
=
F - �v m0
F m0 = F m0 + �tF - m0�v - �2vt
(m0 + �t)v = F t
F t
v =
m0 + �t
�dt
m = m0 - �t
F dt F
= dv �! v = - ln(m0 - �t)|t
0
m - �t �
F m0
v = ln
� m0 - �t
ńł
�łQ - N = m2a
�ł
2
�ł
ółN - Q1 = m1a
m2 - m1
Q2 - Q1 = (m1 + m2)a �! a = g
m1 + m2
-m2 + m1m2 + m2 + m1m2 2m1m2
1 1
N = m1a + Q1 = g = g
m1 + m2 m1 + m2
4m1m2
2N = g
m1 + m2
�g = 2N
4m1m2
� =
m1 + m2
 0 d" T d" fsN
Tk = fkN
fk d" fs
 S S au
�
a = a + au + 2� � v + � � (� � r )
ma = F
m [a + au + 2� � v + � � (� � r )] = F
ma = F -mak -2m� � v -m� � (� � r )
Fc Fod
�
d�
ma = F - mak - m � r - 2m� � v - m� � (� � r )
dt
ma = F + FB
FB
v = -� � r
Fc = -2m� � v = -2m� � (-� � r ) = 2m� � (� � r )
Fod = -m� � (� � r )
F� = Fc - Fod = 2m� � (� � r ) - � � (� � r ) = � � (� � r )

FC = -2m� � v = 2mv � �
"
"
�n = � sin �, �t = �cos�
Fc = 2mv � � = 2mv � �n
�n
2Ą 2Ą
T = = H"
�n � sin �
"
m = 70 v = 6, 5 �! Fc <" 1, 5 � 10-2
m = 1, 5 v = 100 �! Fc <" 5
<"
"
"
"
" �t
 W = F � s = F s cos ą
B
WAB = F (s) � ds
A
" F (s)
"
" FĄ"
dr = d� � r
dW = F � (d� � r) = (d� � r) � F = d� � (r � F )
F
dv
F = m � a = m
dt
B tB tB tB
dv
WAB = F ds = F vdt = m vdt = m vdv =
dt
A tA tA tA
vB
2 2
mv2 mvB mvA
= = -
2 2 2
vA
mv2
2
mv2 p2
Ek = =
2 2m
WAB = Ek(B) - Ek(A)
B hB
WAB = F ds = mg � dh = mg(hB - hA) = mgh
A hA
h
Ep = mgh
F = kx
xB
B B B
kx2
WAB = F ds = F dx = kxdx =
2
A A A
xA
kx2 kx2
B A
WAB = -
2 2
kx2
Eps =
2
dW F ds Md�
P = = = F � v P = = M�
dt dt dt
WAB F F
r r
Ep(r) = - F dr + Ep(r0) = - F dr + const
r0 r0
dEp = -F dr = -(Fxdx + Fydy + Fzdz)
"Ep "Ep "Ep
Fx = - , Fy = - , Fz = -
"x "y "z
" " "
" a" ( , , )
"x "y "z
"Ep "Ep "Ep
F = -"Ep = -( , , ) a" - Ep
"x "y "z
(Ek + Ep = const)
dp dp
F = F = 0 �! = 0 �! p = const
dt dt
"
"
t
t + "t
M � v = (M - "m)(v + "v) + "m(v - u)
M � v = M � v + M � "v - v � "m - "m � "v + v � "m - u � "m
M � "v = u � "m "m = � � "t
M = M0 - �t �
M"v = � � u � "t
"v
M = �u �! Fc = Ma = �u = const
"t
=a
dv
(M0 - �t) = �u
dt
dt dv
= � u
M0 - �t �u
t v
-�dt dv
- =
M0 - �t u
0 0
t v
v
- ln(M0 - �t) =
0 u 0
M0 M0
v = u ln = u ln
M0 - �t M
m1r1 + m2r2 + . . . miri
RCM = =
m1 + m2 + . . . mi
mivi pi
VCM = =
mi mi
pi = mi � VCM = MC � VCM
piCM = 0
 M = r � F
L = r � p
dL
=?
dt
dL d dr dp
= (r � p) = � p +r �
dt dt dt dt
=0
dr
� p = v � p = v � (mv) = 0
dt
dL dp
= r � = r � F = M
dt dt
dL
= M
dt
dL
M = 0 �! = 0 �! L = const
dt
L = r � p, p = mv = m� � r
L = mr � (� � r)
A � (B � C) = B(A � C) - C(A � B)
L = m[�(r2) - r(�r)]
2
L = mi[�ri - ri(ri�)]
ri� = xi�x + yi�y + zi�z
2
Lx = �x mi(ri - x2) -�y mixiyi -�z mixizi
i
Ixx -Ixy -Ixz
2
Ixx = m(ri - x2)
i
Ixy = - mixiyi
Ixz = - mixizi
 Iy..., Iz...
Lx = Ixx�x + Ixy�y + Ixz�z
Ly = Iyx�x + Iyy�y + Iyz�z
Lz = Izx�x + Izy�y + Izz�z
L = I � �
�ł �ł
Ixx Ixy Ixz
�ł �ł
�ł �ł
I =
�ł �ł
�łIyx Iyy Iyzłł
Izx Izy Izz
Ixy = Iyx Ixz = Izx Iyz = Izy
Ią� = �(r)(r2 � �ą� - xąx�)dv
m
" m l  =
l
l l
r3 l3 ml2
I = r2dm = r2dr =  r2dr =  |l = =
0
3 3 3
0 0
l/2
r3 l3 l3
I0 = r2dr =  |l/2 = +
-l/2
3 24 24
-l/2
ml2
I0 =
12
2
ml2 ml2 ml2 l
I - I0 = - = = m
3 12 4 2
I = I0 + md2
"
mr2
I0 =
2
"
I0 = mr2
"
2
I0 = mr2
5
"
ml2
I0 =
12
dL L
dW = Md� M = � =
dt I
dL d� L L2 I�2
dW = d� = dL = �dL = dL �! Ek = =
dt dl I 2I 2
M = 0 �! L = const �! I� = const
"
L �
dL Ą" L
LĄ" = L sin ą
�p
d d
= + �p�
dt dt
dL
= �p � L
dt
M = �p � L
mgr sin ą = �pL sin ą
mgr mgr
�p = =
L I�
Kp
dL
= M
dt
dL dL
= �p � L +
dt dt
=0
dL
- �p � L = 0
dt
Kp
M
L
Kp = -�p � L = L � �p
I0 = mRs
R
T = 2Ą H"
g
84
2Ą 2ĄI�
T = =
�p mga
�
23ć%27
27725 46.79
9.2 19
rad
�p = 46.79 /rok = 7.19 � 10-12
s
F �l
= E
s l
s
�l
E
F = k�l
F = -kx
d2x d2x k
m = -kx �! = - x
dt2 dt2 m
k
2
�0 a"
m
2
ć = -�0x
x = A sin(�0t + �)
2Ą 2Ą m
�0 = = 2Ą� T = = 2Ą
T �0 k
x
kx2
Ep = - F dx = �!-
2
0
"
x
F = mg sin � H" mg
l
mg
F = - x
l
=k
m l
T = 2Ą = 2Ą
k g
"
M = d � Q
M = mgd sin ą H" mgdą
d2ą
M = - mgd ą I = M
dt2
k
I I
T = 2Ą = 2Ą
k mgd
1
Ec = Ek + Ep = const. = kA2
2
dx
FT = -bv = -b
dt
d2x dx
m = -kx - b
dt2 dt
d2x k b dx
= - x -
dt2 m m dt
b k
2
a" 2�, �0 a"
m m
2
ć + 2�� + �0x = 0
x = Ae-�t cos(�t + �)
2
� = �0 - �2
2
�0 - �2 = 0
xm(t) 2Ą�
ln = �T =
xm(t + T ) �
Fw = Fm sin(&!t)
d2x dx
m = -kx - b + Fm sin(&!t)
dt2 dt
Fm
2
ć + 2�� + �0x = sin(&!t)
m
x = A sin(&!t + �)
Fm
A =
2
m (�0 - &!2) + 4�2&!2
2�&!
� = arctan
2
&!2 - �0
2
1 Fm 2�&!2
P =
2
2 m (�0 - &!2) + 4�2&!2
&! - �0
�1 - �2 �1 + �2
x = A cos(�1t) + A cos(�2t) = 2A cos t � cos t
2 2
1
�
�D
2
2Ą 2Ą 1
T0 = = =
�D �1 - �2 �1 - �2
�D = �� = �1 - �2
Ą
�1 = �2 (� = 0, Ą), (� = ),
2
Tk = 23, 32 = 2, 36 � 106s
RZK = 3, 84 � 108m H" 60Rz
2
2Ą m 9, 80 g
ad = �2RZK = RZK = 2, 72 � 10-3 H" =
T s2 602 (RZK/RZ)2
Mm r
F = -G �
r2 r
 t
m � dM
dF = dF1 � cos ą = G cos ą
x2
dM = �dV = �2Ąr sin �rd�t = 2Ąt�r2 sin �d�
R - r cos �
cosą =
x
R2 + r2 - x2
r cos � =
2R
2xdx x
-r sin �d� = - �! sin �d� = dx
2R Rr
ĄGt�mr R2 - r2
dF = + 1 dx
R2 x2
ą
R+r
R2 - r2 R2 - r2 R+r
F = ą + 1 dx = ą + x =
x2 -x
R-r
R-r
R2 - r2 R2 - r2
= ą - + R + r + - R + r =
R + r R - r
= ą [-R + r + R + r + R + r - R + r] = 4ąr
4Ąr2�tm Mm
F = G = G
R2 R2
r+R
R2 - r2 R2 - r2 r+R
F = ą + 1 dx = ą + x =
x2 -x
r-R
r-R
= ą[-R + r + R + r - R - r - r + R] = 0
r
r
Mm Mm Mm
Ep = W"-r = G dr = -G = -G
r2 r r
"
"
"Ep Mm
F = -gradEp = - r = -G r
Ć Ć
"r r2
M r 4
Ąr3��4Ąr2
Gmdm
3
W = - = - G dr =
r r
0 0
r
G(4Ąr3�)2
16 16 r5 3
3
= - G Ą2�2r4dr = -G Ą2�2 = -
3 3 5 5 r
0
3 GM2
W = -
5 r
d2r1 GM1M2
M1 = - r
Ć
dt2 r2
d2r2 GM1M2
M2 = r
Ć
dt2 r2
M1r1 + M2r2
RCM =
M1 + M2
d2r1 GM2 d2r2 GM1
= - r; = r
Ć Ć
dt2 r2 dt2 r2
Ł
� �
M1r1 + M2r2 = 0 �! M1rŁ1 + M2rŁ2 = const �! RCM = const
= 0
d2r d2(r1 - r2) 1 1 GM1M2
= = - + r
Ć
dt2 dt2 M1 M2 r2
1 1 1
a" +
� M1 M2
d2r GM1M2
� = - r
Ć
dt2 r2
M = 0 �! L = const
l = r � �v = const �! Ą" L
v = Y�r + r�u�
Ł Ć
L = r � �v = �r � (Y�r + r��p) = �r2��z
Ł Ł
�v2 �(Y2 + r2�2) �Y2 �r2�2
Ł Ł
Ek = = = + =
2 2�r2 2 2
�Y2 (�r2�)2 �Y2 L2
Ł
= + = +
2 2 2 2�r2
L2
2�r2
L2 GM1M2 L2
Eef = Ep + = - +
2�r2 r 2�r2
GM1M2
�� = - + ��2r
r
r2
" E < 0
" E = 0
" E > 0
1
"S = r � "r
2
dS 1 dr 1 L
= r � = r � p =
dt 2 dt 2m 2m
dS
L = const �! = const
dt
1 1
= (1 - e cos �)
r se
s
e
Mm 2Ą
G = m�2r � =
r2 T
2
T 4Ą2
=
r3 GM
4Ą2a3
2
T =
G(M1 + M2)
C c
Ep = - F = - r = Ma
Ć
r r2
Ł
rF = Mrv
rMa
c
Ł
- = Ep = Mrv
r
d
Ł Ł
M (rv) = MYv + Mrv = Mv2 +Mrv
dt
2Ek
d
M (rv) = 2Ek + Ep
dt
 rv
< 2Ek >= - < Ep >
Ep
Ek
F N
p = 1P a = 1
S m2
kG
m2
1 dv
B = - |E=const
V dp
1
� =
B
" 50cm3 50cm3 96cm3
3, 7%
N
5, 4 � 108
m2
"
"
dp = �gdh
ph = �gh
Sv� = const
S
v
�
V (x) = vx"y"z"t
"vx
V (x + "x) = vx + "x "y"z"t
"x
"vx
"V = V (x + "x) - V (x) = "x"y"z"t
"x
"vx "vy "vz
+ + = "v = v
"x "y "z
v = 0
div(�v) = 0
"�
(�v) + = S
"t
"�
"t
S
�v2
�gh + + p = const
2
p"V
mgh = �"V gh
mv2 �"V v2
=
2 2
dv
� = �f - grad p
dt
f
"
"
"
"
"
"
"
"
"
Ft = -6Ą�rv
�
r�v
Re =
�
�v2
Fc = C� S
2
C�
�v2
2
C�
Re < 1160
Re > 1160
� = vds
vds
v = lim
A0
A
v = 0

x w
ę
Ć
" " "
rotv = " � v =
"x "y "z
vx vy vz
F
 =
2l
Ąr2p rp
 = =
2Ąr 2
2
p = �!-
r
 Tt = 273, 16K
t[ć%C] = T [K] - 273, 15ć%
5ć%
1ć%F = C
9
-40ć%C = -40ć%F
100ć%C = 212ć%F
0ć%C = 32ć%F
"U = W + Q
U
dU = �W + �Q
W = F dx = p Sdx = pdV
dV
�W = -pdV
�Q = mcwdT
cw
�Q
dS =
T
"S e" 0
dU = T dS - pdV
"
"
U 3
= kT
N 2
J
<"
k 1, 38 � 10-23
=
K
"
dv dv
m = F � r �! m r = F � r
dt dt
d dr dv dv
(r � v) = v + r = v2 + r
dt dt dt dt
dv d
r = (r � v) - v2
dt dt
d
m (r � v) = F � r + mv2
dt
m d F � r
(r � v) = + Ek
2 dt 2
1
< Ek >= - < F � r >
2
F r F � r = 0
F � r = -F l = -(pl2)l = -pl3 = -pV
< F � r >= -3pV
3
< Ek >= pV
2
< Ek >= U
2
pV = U
3
3
U = kT N
2
pV
= nR
T
m pV m pV R
n = = R =
� T � T m �
p R
= = const
�T �
 (T = const) pV = const
U = const
Vb Vb
const Vb Va
W = - pdV = - dV = -const ln = nRT ln
V Va Vb
Va Va
W + Q = 0 �! Q = -W
(p = const)
V nR
= = const
T p
Vb
W = - pdV = -p(Vb - Va) = p(Va - Vb)
Va
Q = mcwp(Tb - Ta) = n �cp (Tb - Ta)
cmol
p
cmol = �cp
p
 (V = const)
p �R
= = const
T V
W = 0 Q = mcv(Tb - Ta)
"U = Q
mcvdT = -pdV + mcxdT
p dV dV nR
p = const : cv = - + cp =
m dT dT p
p
R
cp = cv +
�
cmol = cmol + R
p v
�Q = 0
dU = �W mcvdT = -pdV
m RT
mcvdT = dV
� V
�cv dT dV �cv
= - �! ln T + ln V = const
R T V R
�cv �cv
R R
T V = const �! (pV ) V = const
cmol
�cv + R
p
� = =
�cv cmol
v
cp
= � > 1
cv
�
pV = const
1
kT
2
1
E = (p2 + p2 + p2)
x y z
2m
3 1 "U 3 3 5
U = NkT cmol = = R cmol = R + R = R
v p
2 n "T 2 2 2
v
cp 5
� = = = 1, 67
cv 3
1 1
2 2
E = (p2 + p2 + p2) + (l1 + l2)
x y z
2m 2I
5 5 7 7
U = NkT cmol = R cmol = R � = = 1, 4
v p
2 2 2 5
6 6 4
U = NkT cmol = R cmol = 4R � = = 1, 33
v p
2 2 3
1 1
E = p2 + ąx2
x
2m 2
U = NkT
cmol = R
v
J
cmol = 3R H" 25
v
mol � K
3R
W = Q
Q
"S = - < 0
T
W = Q - Q
Q Q
"S = "S + "S = - + e" 0
T T
Q Q - W
"S = - + e" 0
T T
W Q Q W T T - T
d" - �! d" 1 - =
T T T Q T T
W T - T
� a" d"
Q T
Q - Q Q T (S - S ) T
� = = 1 - = 1 - = 1 -
Q Q T (S - S ) T
p
T = const = const dp = -�gdh
�
p p0 p
= �! � = �0
� �0 p0
p
dp = - �0gdh
p0
dp �0
= - gdh
p p0
�0
ln p - ln p0 = - gh
p0
�0g
p = p0 exp - h
p0
�Q = T dS, �W = -pdV, dU = mcvdT
dU = �Q + �W
p m R
mcvdT = T dS - pdV =
T � V
dT p
dS = mcv + dV
T T
m
T2 V2
"S = mcv ln T + R ln V
T1 V1
�
T2 mR V2
S2 - S1 = mcv ln + ln
T1 � V1
2
N
p + a (V - Nb) = kNT
V
"
"
"
"
"
" y = f(x - �t)
" y = f(x + �t)
1
�
T
 vT
2Ą
� = 2Ą�
T
2Ą
k =

x t
y = f - = f(kx - �t)
 T
 �
vf = =
T k
dvf
d
d� �
vgr = ; vf =
dk k
d dvf 2Ą dvf d
vgr = (kvf) = vf + k = vf +
dk dk  d dk
d d 2Ą 2Ą 
= = - = -
dk dk k k2 2Ą
dvf
vgr = vf - 
d
vgr = vf

Fw = F
"m = �"l, � F
"mv2
2F sin � =
R
"l/2 �"lv2
2F =
R R
F
v =
�
F �
� = � =
S S
�
v =
�
Fw = S�(x + "x) sin ą(x + "x) - S�(x) sin ą(x)
"y
sin ą(x) H" tan ą(x) =
"x
�(x + "x) H" �(x)
"y "y "y "2y "y "2y
Fw = S� (x + "x) - (x) = S� (x) + "x - (x) = S� "x
"x "x "x "x2 "x "x2
"2y "2y
�S"x = S� "x
"t2 "x2
"2y � "2y "2y 1 "2y
= ; =
"x2 � "t2 "x2 v2 "t2
� F
v = =
� �
�(x, t)
x x + "x
"2�
�S"x = S�(x + "x) - S�(x)
"t2
"2� �(x + "x) - �(x) "�
� = =
"t2 "x "x
"�
� = E� � =
"x
"2� E "2� E
= �! v =
"t2 � "x2 �
Ć
v =
�
Ć
1
ł = �t �t = Ćł
Ć
Ć
"F "l
= -E
S l
"F "l "V
= "p; =
S l V
"V dp
"p = -E �! E = -V a" �, �
V dV
E �
v = =
� �
� � �-1
pV = const V dp + pV �dV = 0
dp
� = - V = �p
dV
�p p R
v = =
� �T �
�RT
v =
�

v
u
t1 = 1s
t1
s1 = ut1 = uT = �0
T

s1 �0
v v v
s = vt1 = ut1 = �0 �
u u u
v
� = s = s + s1 = �0 1 +
u
�0v v
� = �0 + = �0 1 +
u u
�0v v
� = �0 - = �0 1 -
u u
a
a
t1 =
u
 t = T0 (a - vT0)
a - vT0
t2 = + T0
u
a - vT0 a v
T = t2 - t1 = + T0 - = T0 1 -
u u u
1 1 �0
� = = =
v
v
T
1 -
T0 1 -
u
u
a + vT0 v
t2 = + T0 T = t2 - t1 = T0 1 +
u u
�0
� =
v
1 +
u
y = F (x + ut) - F (-x + ut)
x t x t
y = A sin 2Ą + - A sin 2Ą - + =
 T  T
x t x t
= A sin 2Ą + - sin 2Ą - + =
 T  T
x t
= 2A sin 2Ą cos 2Ą
 T
x  
2Ą = nĄ �! x = n = 2l
 2 4

x = (2l + 1)
4
� >
� <
�RT
v =
�
m
0ć%C - v = 331, 8
sm
20ć%C - v = 343, 8
sm
100ć%C - v = 387, 2
s
W
-I0 = 10-12
m2
I
� = 10 log
I0
km
c = 214300
s
ł 40, 5 =
2ą
vz
tan ą = �! ą H" 20, 5
c
km
c = 315300 ą 500
s
km
c = 289000 ą 500
s
km
c = 299796 ą 4
s
c = � � = 5960, 9000, 9500
km
c = 299792, 5 ą 1
s
km
c = 299794, 2 ą 1, 9
s
km
c = 299793, 1 ą 0, 3
s
m
c = (2, 997925 ą 0, 000003) � 108
s
m
c = 299792458
s
1s a" 9192631770
L1 L1 2L1c
vz : t1 = + =
2
c - vz c + vz c2 - vz
s 2 L2 + x2
2
Ą" vz : t2 = =
c c
vz
L2
x vz L2
2
c
" = �! x = ; L2 + c2 =
2
2
2
vz
L2 + x2 c
vz
1 -
1 -
c2
c2
2L2 1
t2 =
2
c
vz
1 -
c2
2L1c 2L2 1
"t = t1 - t2 = -
2
2
c2 - vz c
vz
1 -
c2
90ć% (L1 �! L2)
2L2c 2L1 1
"t = t 1 - t 2 = -
2
2
c2 - vz c
vz
1 -
c2
�ł �ł
�ł �ł
2
2 1 vz
�ł
� = "t + "t = (L1 + L2) - 1�ł H" (L1 + L2)
�ł �ł
2 2
�ł
vz vz łł c3
c 1 - 1 -
c2 c2
c� Vz2 L1 + L2
k = H"
 c2 
km
vz = 30 ,  H" 6�10-7m, L1 = L2 = 1, 2m
s
k H" 0, 04
x = x - vt
y = y, z = z, t = t
x = Ax + B
ńł
�ł
�łx
�ł
= ł(x - vt)
�ł
�ł
�ł
�ł
�ł
�łt = ł(t - ąx)
�ł
�ł
�ły
�ł
= y
�ł
�ł
�ł
�ł
�ł
�łz = z
ół
ł1, ł2, ą
x2 + y2 + z2 = ct2
x 2 + y 2 + z 2 = ct 2
2 2
ł1(x2 - 2vxt + v2t2) + y2 + z2 = c2ł2(t2 - 2ątx + ą2x2)
v2
2 2 2 2 2 2
x2 (ł1 - ł2c2ą2) +2xt [-ł1v + c2ł2ą] +y2 + z2 = c2t2 [ł2 - ł1 ]
c2
1 0
1
ńł
�ł
2 2
�łł1 - ł2c2ą2 = 1
�ł
�ł
�ł
�ł
c2ą
2 2 2 2
�ł-ł1v + c2ł2ą = 0 �! ł1 = v ł2
�ł
�ł
�ł
�ł
2 2
ółł2 - ł1 v2
= 1
c2
ńł
�ł
�łc2ął - ł2c2ą2 = 1 �! ł2 c2ą - c2ą2 = 1
2 2 2
�ł
�ł
2
�ł
v v
�ł
�ł
c2ą v2
2 2 2
2
�łł - ł2 v c2 = 1 �! ł2(1 - ąv) = 1
�ł
�ł
�ł
�ł
�ł
ół
c2ą
- c2ą2 = 1 - ąv
v
c2
-c2ą2 + ą + v - 1 = 0
v
2 2
c2 c4 c2
" = + v - 4c2 = + 2c2 + v2 - 4c2 = - v
v v2 v
"
c2
" = - v c > v
v
c2 c2
- - v - + v
1
v v
ą1 = =
-2c2 v
c2 c2
- - v + - v
v
v v
ą2 = =
-2c2 c2
1
2 2
ł2 = �! ł1(ą1) = "
1 - ąv
1 1
2
ł2(ą2) = =
v
v2
1 - v
c2 1 - c2
v
c2 1
c2ą
2 2 c2 2
ł1 = ł2 = = ł2
v v v2
1 -
c2
v 1
2 2
ą = , ł1 = ł2 = ł2 =
c2 v2
1 -
c2
v 1 1
� = , ł = = > 1
c
1 - �2
v2
1 -
c2
ńł
�ł
�łx
�ł
= ł(x - vt) = ł(x - �ct)
�ł
�ł
�ł
�ł
�ł
�ły = y
�ł
�ł
�łz
�ł
= z
�ł
�ł
�ł
�ł
�ł
�łt = ł(t - v x) = ł(t - � x)
ół
c2 c
ńł
�ł
�łx = ł(x + �ct)
�ł
�ł
�ł
�ł
�ł
�ł
�ły = y
�ł
�ł
�łz = z
�ł
�ł
�ł
�ł
�ł
�ł
�łt = ł(t + � x )
ół
c
 x , y , z , t
�
x 2 + y 2 + z 2 - c2t 2 = ł2(x - �ct)2 + y2 + z2 - c2ł2(t - x)2 =
c
= ł2x2 - 2ł2�cxt + ł2(�ct)2 + y2 + z2 - c2ł2t2 + 2ł2�cxt - ł2�2x2 =
x2 (1 - �2)ł2 +y2 + z2 - c2t ł2(1 - �2) = x2 + y2 + z2 - c2t2
1 1
x2 + y2 + z2 - c2t2 = const
S = c2"t2 - "x2 - "y2 - "z2 = S
S > 0
S = 0
S < 0
�
dx = łdx + ł�xdt ; dt = łdt + ł dx ; dy = dy
c
dx łdx + ł�cdt vx + �c vx + v
vx = = = =
�
vx� vxv
dt
łdt + ł dx
1 + 1 +
c
c c2
1
2
vy
dy dy v2
vy = = = 1 -
�
vxv
dt c2
łdt + ł dx
1 +
c
c2
vx - v vx
vx = - 1 - vx - v
vxv
c
1 -
c2
dvx
a x =
dt
�ł �ł
vxv v
1 - - (vx - v) -
d vx - v
c2 c2 dvx =
�ł łł
dvx = dvx =
vxv 2
vxv
dv
1 -
1 -
c2
c2
vxv vxv v2 v2
1 - + - 1 -
dvx
c2 c2 c2 dvx = c2
=
2
vxv 2 vxv 2dvx = vxv
1 - 1 - ł2 1 -
c2 c2 c2
dvx 1 ax
a x = =
2
vxv vxv 3
�
ł2 1 - ł3 1 -
łdt - łdx
c2 c2
c
a x
ax =
3
vxv
ł3 1 +
c2
a x
(vx = 0) �! ax =
ł2
v - c �! ł - " �! ax - 0
"
" L0 = x 2(t 2) - x 1(t 1)
t 1 t 2
"
L = x2(t2) - x1(t1) t2 = t1 = t
L0 = x 2(t 2)-x 1(t 1) = ł(x2-�ct2)-ł(x1-�ct1) = ł(x2-x1)-ł�c (t2 - t1) = łL
0
L0 L0 L1 ł1
L1 = i L2 = �! =
ł1 ł2 L2 ł2
x t �
�
t = ł(t + x ) = ł�
c
t = ł� �
t 1 = 0 (x = 0)
t1 = 0, x1 = 0
t 2 = T0 (x = 0)
v
t2 = ł(t 2 + x 2) = łT0
c2
x2 = ł(x 2 + vt 2) = łvT0
x2
"t = x2
c
łvT0 v
T = t2 + "t = łT0 + = łT0(1 + ) =
c c
v v
1 + 1 +
1 + �
c c
= T0 = T0 = T0
v
1 - �
v2
1 -
1 -
c
c2
1 1 - � 1 + �
� = = �0 ;  = 0
T 1 + � 1 - �
x
cos ą =
ct
x
cos ą =
ct
x v
ł(x + vt ct + c cos ą + �
cos ą = = =
v
v x
1 + � cos ą
cł(t + x )
1 +
c2
c ct
cos ą + 1
� - 1 �! cos ą - = 1 �! ą - 0
1 + cos ą
vy v2 1/2
vy = 1 -
vxv
c2
1 -
c2
"y "y "t
=
"� "t "�
"y "y
= ł = vył
"� "t
mv
p a" = mc�ł = m(v)v
v2 1/2
1 -
c2
1
ł2 = �! ł2 - �2ł2 = 1 � m2c4
0
1 - �2
m2c4ł2 - m2c4�2ł2 = m0c4
0 0
m2c4 - p2c2 = m0c4
1 1 1
mc2 = m0c2 H" m0c2 1 + �2 + . . . = m0c2 + m0v2
2 2
1 - �2
Ek
m0c2
E a" mc2 =
v2
1 -
c2
E2 - p2c2 = m2c4
0
�ł �ł
�ł �ł
dp d m0v
�ł �ł
F = =
�ł �ł
dt dt �ł
v2 łł
1 -
c2
dp
FT =
dt
dv v2 1 v dv
�ł �ł
m0 1 - + m0v
dt c2
v2 c2 dt
�ł �ł 1 -
d m0v
�ł �ł
c2
FT = = =
�ł �ł
dt �ł v2
v2 łł
1 -
1 -
c2
c2
m0 dv m0
= = aT
v2 3/2 dt v2 3/2
1 - 1 -
c2 c2
FN
m0
FN = aN
v2
1 -
c2
F = ma

v FT FN
c mcaT m0aN
E = m0c2 = const
�E
px = ł p x + �! dpx = łdp x
c
dt = łdt , dpy = dp y, dpz = dp z
dpx łdp x dp x
Fx = = = = Fx
dt łdt dt
1 1
Fy = Fy; Fz = Fz
ł ł
x2 x2
dp
W = F dx = dx
dt
x1 x1
dv
m0
dp d m0v
dt
= =
dt dt
v2
v2 3/2
1 -
1 -
c2
c2
dv
t m0 � vdt v
v m0c2
dt
W = = m0 dv = - m0c2 = Ek
ł3
0 0 v2
v2 3/2
1 -
1 -
c2
c2