9 :;0AA.
1. 5@5945< 2 A8AB5<C >BAG5B0,
A2O70==CN A :>@01;5< .  MB>9
A8AB5<5 :>@01;L  42865BAO A
>B=>A8B5;L=>9 A:>@>ABLN
r r r
V = V - V . >4C;L MB>9
>B= 2 1
A:>@>AB8 @025=
r
ą
V = 2v cos , (1)
>B=
2
ą
0 55 25:B>@ =0?@02;5= ?>4 C3;>< = 30� : >B@57:C  (A< @8A).
2
!;54>20B5;L=>, :>@01;L  42865BAO >B=>A8B5;L=> :>@01;O  ?>
?@O<>9 !.
0) 8=8<0;L=>9 @0AAB>O=85 <564C :>@01;O<8 5ABL @0AAB>O=85 >B
B>G:8  4> ?@O<>9 !, :>B>@>5 @02=>
ą L
l = Lsin = . (2)
min
2 2
1) G5284=>, GB> H;N?:0, A?CI5==0O A :>@01;O , 4>AB83=5B
:>@01;O  70 <8=8<0;L=>5 2@5<O, 5A;8 A:>@>ABL 8E A1;865=8O
<0:A8<0;L=0, 0 =0G0;L=>5 @0AAB>O=85 <564C =8<8 <8=8<0;L=>. -B8
CA;>28O 1C4CB 2K?>;=5=K, 5A;8 H;N?:C A@07C A?CAB8BL =0 2>4C 8
=0?@028BL 55 =02AB@5GC :>@01;N . ">340 2@5<O, 70 :>B>@>5 H;N?:0
4>AB83=5B :>@01;O  2KG8A;O5BAO ?> D>@<C;5
L
t = . (3)
min
2v
2) CABL :0?8B0= :>@01;O  >B?@02;O5B
H;N?:C G5@57 2@5<O � (=0<
=5>1E>48<> =09B8 53> <0:A8<0;L=>
2>7<>6=>5 7=0G5=85) 2 B>G:5 S, 0 70B5<
G5@57 2@5<O t H;N?:0 2AB@5G05BAO A
:>@01;5<  2 B>G:5 D (A<. @8A. ). 0 MB>
2@5<O :>@01;L  ?@>945B ?CBL
AD = v t + � . 0: A;54C5B 87 @8A. ,
( )
GB>1K H;N?:0 8 :>@01;L  2AB@5B8;8AL 4>;6=> 2K?>;=OBLAO
CA;>285 (:>B>@>5 A;54C5B 87 B5>@5<K :>A8=CA>2 4;O B@5C3>;L=8:0
BSD)
2
22
ut = v� + L - v t + � - 2 v� L - v t + � cosą . (4)
( ) ( ) ( ) ( ) ( )
() ( )
1
;O B>3>, GB>1K =09B8 <0:A8<0;L=>5 7=0G5=85 2@5<5=8 �
=5>1E>48<> @0AA<>B@5BL 2K@065=85 (4) :0: C@02=5=85 >B=>A8B5;L=>
25;8G8=K t 8 >?@545;8BL CA;>28O (7=0G5=8O � ), ?@8 :>B>@KE >=>
8<55B =5>B@8F0B5;L=>5 @5H5=85.  ?@8=F8?5 MB>B ?CBL @5H5=8O
7040G8 ?@82545B : CA?5EC, ?@0240 ?CB5< 4>;38E 8 3@><>74:8E
0;351@08G5A:8E ?@5>1@07>20=89.
AB0B8, MB> 65 C@02=5=85 (?@8 u = v ) <>6=> 8A?>;L7>20BL 4;O
0;351@08G5A:>3> >1>A=>20=8O @57C;LB0B0, ?>;CG5==>3> 2 ?. 1). 5H82
MB> C@02=5=85 >B=>A8B5;L=> t , <>6=> ?>;CG8BL 7028A8<>ABL
2@5<5=8 42865=8O t + � >B 2@5<5=8 � , 0 70B5< =09B8 <8=8<C<
( )
MB>9 DC=:F88. -B>B A?>A>1 ?@82>48B : C65 ?>;CG5==><C @57C;LB0BC:
DC=:F8O t + � <>=>B>==> 2>7@0AB05B A @>AB>< � , A;54>20B5;L=> 55
( )
<8=8<C< 4>AB8305BAO ?@8 � = 0.
5@=5<AO : @5H5=8N
?C=:B0 2).
?OBL @0AA<>B@8<
42865=85 :>@01;59 2
A8AB5<5 >BAG5B0, A2O70==>9
A :>@01;5< .  MB>9
A8AB5<5 4803@0<<0 ?5@5<5I5=89 :>@01;59 8 H;N?:8 8<55B 284,
r r
ą r
?>:070==K9 =0 @8A. , 745AL >1>7=0G5=> � = , u = V - V -
>B= 2 1
2
A:>@>ABL H;N?:8, >B=>A8B5;L=> :>@01;O . 0 @8AC=:5 284=>, GB>
2@5<O � (8;8 GB> B> 65 A0<>5 ?5@5<5I5=85 V� ) 1C45B <0:A8<0;L=>
?@8 <0:A8<0;L=>< C3;5 ł , <564C =0?@02;5=85< >B=>A8B5;L=>9
r
A:>@>AB8 u>B= 8 >B@57:>< . 0:A8<0;L=>5 7=0G5=85 MB>3> C3;0
<>6=> =09B8, ?>AB@>82 4803@0<<C A:>@>AB59 (@8A. ) .
r
5:B>@ A:>@>AB8 H;N?:8 u <>65B 1KBL =0?@02;5= ?>4
?@>872>;L=K< C3;><, 8=K<8 A;>20<8 53> :>=5F <>65B
@0A?>;030BLAO 2 ;N1>9 B>G:5 =0@8A>20==>9 >:@C6=>AB8. 0: A;54C5B
r
87 @8AC=:0 C3>; ł 1C45B <0:A8<0;5=, 5A;8 25:B>@ u>B= 1C45B
u
:0A0B5;L=K< : MB>9 >:@C6=>AB8. "0:8< >1@07><, sinł = .
max
v
0?8H5< B5>@5<C A8=CA>2 4;O B@5C3>;L=8:0 ABS
2
V�max L
= , (5)
sinł sin Ą - � - ł
()
max max
345 Ą - � - ł - C3>; ASB. 7 2K@065=8O (5) =0E>48<
()
max
L sinł L sinł
max max
�max = �" =�" =
V sin � + ł 2vcos� sin� cosł + sinł cos�
()
max max max
L sinł
max
= �" = (6)
v
sin2� 1 - sin2 ł + sinł 2cos2 �
max max
L 2
= �" .
2
v
v
�# ś#
3 - 1 + 3
ś# ź#
�#
u #
B<5B8<, GB> ?@8
1) u 0 �max 0, B.5. H;N?:C =04> A@07C A?CA:0BL =0 2>4C 8 640BL
?>:0 : =59 ?>4?;K25B 2B>@>9 :>@01;L;
2L
2) ?@8 u = v , :0?8B0= <>65B ?>4>640BL 2 B5G5=88 2@5<5=8 �max = ;
3V
3) ?@8 u > v H;N?:0 <>65B 4>3=0BL :>@01;L ?>A;5 ;N1>3> 2@5<5=8
>6840=8O � .
3) !:>@>ABL A=0@O40 1C45B <8=8<0;L=0, 5A;8 >= ?@>;5B8B
<8=8<0;L=>5 @0AAB>O=85, 1C4CG8 2K?CI5= ?>4 C3;>< 45� :
3>@87>=BC. !;54>20B5;L=> MBC A:>@>ABL <>6=> =09B8 87 C@02=5=8O
2
vmin
= lmin , 8;8
g
Lg
v = .
min
2
3
2. 07;8G8O 2 ?>:070=8OE 2>;LB<5B@>2, 2>7=8:0N 87-70 B>3>, GB> >=8
=5 O2;ONBAO 8450;L=K<8, B> 5ABL 8<5NB :>=5G=>5 A>?@>B82;5=85,
:>B>@>5 <K >1>7=0G8< RV , :>B>@>5 A@02=8<> A A>?@>B82;5=85<
@578AB>@>2 R .
0 AE5<5 C:070=K >1>7=0G5=8O B>:>2, B5:CI8E G5@57
@07;8G=K5 M;5<5=BK AE5<K. A?>;L7CO 70:>=K ?>A;54>20B5;L=>3> 8
?0@0;;5;L=>3> A>548=5=8O, <>6=> 70?8A0BL A;54CNI85 C@02=5=8O
U = I RV
3 3
U = I R + RV . (1)
()
2 3
U = I R + U
1 2 2
K@078< A8;C B>:0 I2 G5@57 A8;C B>:0 I3, 8A?>;L7CO A8AB5<C
C@02=5=89
I = I '+I
2 2 3
, (2)
I ' RV = U
22
87 :>B>@>9 A;54C5B
U
2
I = + I . (3)
2 3
RV
5 A<>B@O =0 B>, GB> 2 A8AB5<5 4 C@02=5=89 (1), (3) A>45@68BAO 5
=58725AB=KE, 87 =55 <>6=> =09B8 7=0G5=85 U .
2
59AB28B5;L=>, 2 B@5BL5 C@02=5=85 A8AB5<K (1) ?>4AB028<
2K@065=85 (3)
�# ś#
U
2
(4)
U = + I R + U .
ś# ź#
1 32
�# R #
V
 87 ?5@2KE 42CE C@02=5=89 MB>9 65 A8AB5<K 2K@078<:
I R = U - U (87 @07=>AB8 MB8E C@02=5=89);
3 2 3
R U
2
= - 1 (87 G0AB=>3> MB8E C@02=5=89);
R U
V 3
8 ?>4AB028< 8E 2 C@02=5=85 (4)
�# ś#
U
2
U = U - 1ź# + U - U + U .
ś#
1 2 2 3 2
�# U #
3
5H5=85 MB>3> :204@0B=>3> C@02=5=8O 8<55B 284
4
2
5U + 4U U - U
3 1 2 3
U = H" 86B .
,
2
2
B@8F0B5;L=K9 :>@5=L <K >B1@>A8;8, :0: =5 8<5NI89 D878G5A:>3>
A<KA;0.
B<5B8<, GB> 2 =0H59 F5?8 RV H" 12R , GB> ?>4B25@6405B =0H5
8AE>4=>5 ?@54?>;>65=85.
3. CABL F8;8=4@ ?>4=O;AO =04 2>4>9 =0 2KA>BC x .
">340 459AB2CNI0O =0 =53> A8;0 @E8<540 @02=0
F = � S h - x g . (1)
( )
A 0
"0: :0: MB0 A8;0 87<5=O5BAO ?> ;8=59=><C 70:>=C,
B> 4;O 2KG8A;5=8O 55 @01>BK <>6=> 8A?>;L7>20BL
55 A@54=55 7=0G5=85. B0:, @01>B0 A8;K @E8<540
1
A = � Shg �" h (2)
A 0
2
?>H;0 =0 C25;8G5=85 :8=5B8G5A:>9 8 ?>B5=F80;L=>9 M=5@388
F8;8=4@0
2
1 �Shv
2
� Sh g = �Shg �" h + . (3)
0
22
7 MB>3> C@02=5=8O >?@545;O5< A:>@>ABL F8;8=4@0
� - 2� <
0
v = gh H" 17 .
,
� A
�
0
1@0B8B5 2=8<0=85, ?@8 � > F8;8=4@ =5 2KA:>G8B 87 2>4K
2
?>;=>ABLN.
4. C45< AG8B0BL, GB> ?@>B5:0O ?> >B>?8B5;L=K< @0480B>@0<, 2>40
>ABK205B 4> :><=0B=>9 B5<?5@0BC@K. ;O B>3>, GB>1K B5<?5@0BC@0 2
:><=0B5 >AB0;0AL =587<5==>9, =5>1E>48<>, GB>1K ?>A;5 @5<>=B0
2>40 ?@8=>A8;0 2 548=8FC 2@5<5=8 B0:>5 65 :>;8G5AB2> B5?;>BK,
GB> 2K@0605BAO C@02=5=85<
c�v S t - t = c�v S t - t .
( ) ( )
1 1 1 0 2 2 2 0
7 MB>3> C@02=5=8O >?@545;O5< A:>@>ABL 42865=8O 2>4K ?> B@C10<
S t - t
( )
1 1 0
v = v .
2 1
S t - t
( )
2 2 0
5H5=85 7040G.
5
10 :;0AA.
1. @8 A<5I5=88 <CDBK =0
@0AAB>O=85 x 28AOI89 3@C7
?>4=8<5BAO =0 2KA>BC
2 2
y = x + h - h . (1)
KG8A;OO ?@>872>4=CN ?> 2@5<5=8 >B
MB>3> 2K@065=8O, CAB0=>28< A2O7L
<564C A:>@>ABO<8 <CDBK v 8 3@C70 u
x
u = v . (2)
2 2
x + h
0<5B8<, GB> ?>A;54=55 A>>B=>H5=85 u = v cosą <>6=> =09B8 ?CB5<
35><5B@8G5A:8E 25:B>@=KE ?>AB@>5=89.
0) 01>B0 2=5H=8E A8; ?@8 A<5I5=88 <CDBK @02=0 87<5=5=8N
?>B5=F80;L=>9 M=5@388 3@C70, ?>MB><C
2
A = mgy = mg x0 + h2 - h (3)
().
1) 0<5B8<, GB> :>340 <CDB0 ?@>E>48B ?>;>65=85 @02=>25A8O (=8BL
25@B8:0;L=0), A:>@>ABL 3@C70 >1@0I05BAO 2 =C;L. >MB><C <CDB0
1C45B 8<5BL <0:A8<0;L=CN A:>@>ABL 8<5==> ?@8 ?@>E>645=88
?>;>65=8O @02=>25A8O, B0: :0: 2 MB>< ?>;>65=88 87<5=5=85
?>B5=F80;L=>9 M=5@388 <0:A8<0;L=>, 8 2AO 70?0A5==0O M=5@38O (3)
?5@5945B 2 :8=5B8G5A:CN M=5@38N <CDBK. -BC <0:A8<0;L=CN
A:>@>ABL =0945< 87 70:>=0 A>E@0=5=8O <5E0=8G5A:>9 M=5@388
2
mv
2 2
max
= mg x + h - h
( 0 ),
2
8;8
2 2
(4)
v = 2g x + h - h
max ().
0
2) ;O >?@545;5=8O A:>@>AB8 <CDBK 2 ?@>872>;L=>9 B>G:5 >?OBL
2>A?>;L7C5<AO 70:>=>< A>E@0=5=8O <5E0=8G5A:>9 M=5@388
(:8=5B8G5A:0O M=5@38O <CDBK 8 3@C70 @02=0 87<5=5=8N
?>B5=F80;L=>9 M=5@388 3@C70):
2 2
mv mu
2 2 2 2
+ = mg x + h - h x + h - h .
()-()
( 0 )
2 2
A?>;L7CO A>>B=>H5=85 (2), =0E>48< 8A:><K5 A:>@>AB8
2 2
x + h
2 2 2 2
<CDBK v = 2g x + h - x + h , (5)
()
0
2 2
2x + h
2
x
2 2 2 2
8 3@C70 u = 2g x + h - x + h (6)
().
0
2 2
2x + h
6
;O ?>AB@>5=8O 3@0D8:>2 MB8E DC=:F89 8E C4>1=> ?@54AB028BL 2
2845
2
v � + 1
22
= � + 1 - � + 1
();,
0
2
2gh 2� + 1
2
u �
22
= � + 1 - � + 1
();
0
2
2gh 2� + 1
x
345 >1>7=0G5=> � = . @0D8:8 <>4C;59 MB8E DC=:F89 (?@8
h
�0 = 2, �0 = 4 ) ?@54AB02;5=K =0 @8AC=:5.
3) 1@0B8< 2=8<0=85, GB> G8A;5==K5 7=0G5=8O ?0@0<5B@>2 B0:>2K,
GB> x0 >> h. >MB><C ?@0:B8G5A:8 2A5 2@5<O 42865=8O (70
8A:;NG5=85< <0;>3> CG0AB:0 21;878 ?>;>65=8O @02=>25A8O) =8BL,
C45@6820NI0O <CDBC, 3>@87>=B0;L=0.  MB>< A;CG05 <>6=>
?@81;865==> AG8B0BL, GB> <CDB0 42865BAO A ?>AB>O==K<
g
CA:>@5=85< a = (C1548B5AL 2 MB>< A0<>AB>OB5;L=>).
2
!;54>20B5;L=>, 2@5<O 55 42865=8O >B :@09=53> ?>;>65=8O 4>
2x x
00
?>;>65=8O @02=>25A8O >?@545;O5BAO D>@<C;>9 � == 2 , 0
a g
x
0
?5@8>4 42865=8O, >G5284=> 2 G5BK@5 @070 1>;LH5 T = 8 H" 2,5c .
g
2. @8 =5?>4286=>9 =0:;>==>9 ?;>A:>AB8 A:>;L65=85 1@CA:0
=0G8=05BAO :>340 ?@>5:F8O A8;K BO65AB8 =0 =0:;>==CN ?;>A:>ABL
?@52KH05B <0:A8<0;L=CN A8;C B@5=8O ?>:>O, :0: 8725AB=> MB>
7
3@0=8G=>5 CA;>285 A2O7K205B C3>; =0:;>=0 8 :>MDD8F85=B B@5=8O
A>>B=>H5=85<
� = tgą . (1)
@8 @02=><5@=>< 2@0I5=88 ?;>A:>AB8
H0910 42865BAO A F5=B@>AB@5<8B5;L=K<
CA:>@5=85< a = &!2l , ?>MB><C 2 ?@>5:F88
=0 =0:;>==CN ?;>A:>ABL C@02=5=85
2B>@>3> 70:>=0 LNB>=0 1C45B 8<5BL 284
(<K ?@54?>;0305<, GB> H0910 AB@5<8BAO
A>A:>;L7=CBL 2=87):
m&!2l = mg sin� - FB@. (2)
!:>;L65=85 =0G=5BAO, :>340 F 4>AB83=5B 25;8G8=K
B@.
�N = �mg cos� . (3)
7 C@02=5=89 (1)-(3) =0E>48<
sin �
g g ( - ą)
&!= sin� - tgą �" cos� = (4)
()
l l cosą
0<5B8<, GB> ?@8 1>;LH8E C3;>2KE A:>@>ABOE H0910 <>65B =0G0BL
A:>;L78BL 225@E ?> =0:;>==>9 ?;>A:>AB8, 2 MB>< A;CG05 A8;0 B@5=8O
87<5=8B =0?@02;5=85 =0 ?@>B82>?>;>6=>5. "0:>5 42865=85
=0G=5BAO, 5A;8 C3;>20O A:>@>ABL 4>AB83=5B 25;8G8=K
sin � +
g g ( ą)
&! = sin� + tgą �" cos� = . (5)
()
2
l l cosą
"0: :0: 2 CA;>288 7040G8, =5 C:070=> =0?@02;5=85 A42830 H091K, B>
40==0O 7040G0 8<55B 420 >B25B0 (4) 8 (5).
3. 02;5=85 684:>AB8 =0 4=> A>AC40 <>65B
8AG57=CBL, 5A;8 ?>4 459AB285< ?@8;>65==>3>
=0?@O65=8O 2 684:>AB8 ?>O28BAO B0:>9
M;5:B@8G5A:89 B>:, :>B>@K9 2708<>459AB2CO A
<03=8B=K< ?>;5<, ?@82545B : ?>O2;5=8N A8;K
<?5@0, :>B>@0O :><?5=A8@C5B A8;C BO65AB8.
>=OB=>, GB> B>: 4>;65= B5GL ?5@?5=48:C;O@=>
3@0=O< b � c . K@078< A8;C BO65AB8 8 A8;C
<?5@0 G5@57 ?0@0<5B@K 7040G8
mg = �abcg , (1)
U Ubc Ubc
F = IBa = Ba = Ba = B . (2)
A
R � * a � *
8
@8@02=820O ?>;CG5==K5 2K@065=8O, =0E>48< 8A:><>5 7=0G5=85
�� * ag
=0?@O65=8O U = .
B
4. "0: :0: 70@O4K H0@8:>2 ?@>B82>?>;>6=K, B> H0@8:8 =0G=CB
A1;860BLAO, 2 <><5=B C40@0 ?@>87>945B 8E ?5@570@O4:0, ?>A;5 G53>
H0@8:8 =0G=CB @07J5760BLAO.
!:>@>AB8 H0@8:>2 v 2 <><5=B AB>;:=>25=8O =0945< 87
1
70:>=0 A>E@0=5=8O M=5@388
2
mv q q q q
1 1 2 1 2
2 =- , (1)
2 4Ą� a 4Ą� D
0 0
q q
1 2
745AL - M=5@38O 2708<>459AB28O H0@8:>2, =0E>4OI8EAO =0
4Ą�r
@0AAB>O=88 r . #G8BK20O 70:>= A>E@0=5=8O M;5:B@8G5A:>3> 70@O40 8
@025=AB2> 70@O4>2 H0@8:>2 ?>A;5 AB>;:=>25=8O, ?>;CG8< 25;8G8=C
MB>3> 70@O40
q + q
1 2
q '= q '= . (2)
1 2
2
"0: :0: C40@ H0@8:>2 01A>;NB=> C?@C389, B> 25;8G8=K A:>@>AB59
H0@8:>2 A@07C ?>A;5 AB>;:=>25=8O >AB0=CBAO ?@56=8<8
(5AB5AB25==>, 87<5=OBAO =0?@02;5=8O A:>@>AB59).
0?8H5< >?OBL 70:>= A>E@0=5=8O M=5@388 4;O 42865=8O
H0@8:>2 ?>A;5 AB>;:=>25=8O
2 2
2 2
mv q + q 1 mv q + q 1
�# ś# �# ś#
1 1 2 2 1 2
2 + = 2 + (3)
ś# ź# ś# ź#
�# # �# #
22 4Ą� D 22 4Ą� a
0 0
45 v A:>@>AB8 H0@8:>2 =0E>4OI8EAO =0 @0AAB>O=88 a . 7
2
2K@065=89 (1) 8 (3) <>6=> =09B8 MBC A:>@>ABL.
2
�#�# ś#
1 q + q A<
ś#
1 2
v = ś# - q q H" 10 .
ź# ,
ś# ź#
2 1 2
#
4Ą� Dm 2 A
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