Фізика, збірник задач

jZ^ k jZ^ k2;

2.4.Yd•j _e_dljh^\b]mgZ h[_jlZ}lvky a qZklhlhx n0= h[ k I•key \bfdg_ggy kljmfm yd•j ihqZ\ jmoZlbky j•\ghkih\•evg_gh • ^h amibgdb ajh[b\ N = 1570 h[_jl•\ <bagZqblb dmlh\_ ijbkdhj_ggy ydhjy jZ^ k

2.5. Dhe_kh h[_jlZ}lvky a]•^gh a j•\gyggyf 3 : <W KW2 + Dt3 ^_

: jZ^, < jZ^ k K jZ^ k2 ' jZ^ k3 AgZclb dmlh\m r\b^d•klv ω • dmlh\_ ijbkdhj_ggy ε m fhf_gl qZkm W k,

k_j_^gx dmlh\m r\b^d•klv <ω> • k_j_^g} dmlh\_ ijbkdhj_ggy <ε> aZ ijhf•`hd qZkm \•^ t1 = 2 c ^h t2 = 4 c.

jZ^ k jZ^ k2)

2.6.>bkd jZ^•mkhf 5 kf h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• lZd sh

aZe_`g•klv dmlh\h€ r\b^dhkl• \•^ qZkm aZ^Z}lvky j•\gyggyf ω :W + Dt4, ^_ : jZ^ k2, ' jZ^ k3 <bagZqblb ih\g_ ijbkdhj_ggy .

lhqhd gZ h[h^• ^bkdZ \ fhf_gl qZkm W k i•key ihqZldm jmom • d•evd•klv h[_jl•\ N \bdhgZgbo ^bkdhf f k²; 1,15 h[_jl•\

2.7.Dhe_kh h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• lZd sh dml ch]h

ih\hjhlm aZe_`blv \•^ qZkm yd 3 &W2 ^_ K jZ^ k2 M fhf_gl qZkm W k e•g•cgZ r\b^d•klv lhqdb gZ h[h^• dhe_kZ v = 0,5 f k. AgZclb ih\g_ ijbkdhj_ggy . p•}€ lhqdb f k2)

2.8. L•eh h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• aZ aZdhghf 3	: <W
+ Ct2 ^_ : jZ^, < jZ^ k, K – jZ^ k2 <bagZqblb ih\g_
ijbkdhj_ggy lhqdb ydZ agZoh^blvky gZ \•^klZg• 5	kf \•^ hk•
h[_jlZggy \ fhf_gl qZkm W k. f k2)

2.9.L•eh h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• lZd sh dml ch]h ih\hjhlm

af•gx}lvky aZe_`gh \•^ qZkm t aZ aZdhghf 3		<W – Ct2 ^_ <
= jZ^ k, K	jZ^ k2 <bagZqblb fhf_gl qZkm t3 \ ydbc
l•eh amibgblvky	lZ d•evd•klv h[_jl•\ N l•eZ	^h amibgdb k

h[_jlZ

2.10.L\_j^_ l•eh h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• aZ aZdhghf 3

= <W – Dt3 ^_ < jZ^ k ' jZ^ k3 <bagZqblb k_j_^g• agZq_ggy dmlh\h€ r\b^dhkl• <ω> • dmlh\h]h ijbkdhj_ggy <ε> aZ

ijhf•`hd qZkm \•^ t = 0 ^h amibgdb AgZclb dmlh\_ ijbkdhj_ggy ε \ fhf_gl amibgdb l•eZ jZ^ k jZ^ k2 jZ^ k2)

11

jZ^ k. Ydhx [m^_ dmlh\Zk i•key ihqZldm ]Zevfm-

2.11. Dmlh\Z r\b^d•klv l•eZ af•gx}lvky a]•^gh a j•\gyggyf ω = A + Bt + Ct2, ^_ $ jZ^ k, % jZ^ k2, K jZ^ k3 GZ ydbc dml ih\_jg_lvky

l•eh aZ ijhf•`hd qZkm \•^ t1 = 0 c ^h t2 = 10 c AgZclb k_j_^gx dmlh\m r\b^d•klv <ω> aZ p_c ijhf•`hd qZkm jZ^ jZ^ k

2.12. L\_j^_ l•eh ihqbgZ} h[_jlZlbky gZ\dheh g_jmohfh€ hk• a dmlh\bf ijbkdhj_ggyf ε :W ^_ : jZ^ k3 Q_j_a ydbc qZk t i•key ihqZldm

h[_jlZggy \_dlhj ih\gh]h ijbkdhj_ggy . ^h\•evgh€ lhqdb l•eZ [m^_ ml\hjx\Zlb dml . = 450 a €€ \_dlhjhf r\b^dhkl• v ? k

2.13.L\_j^_ l•eh h[_jlZ}lvky gZ\dheh g_jmohfh€ hk• lZd sh ch]h dmlh\Z r\b^d•klv aZe_`blv \•^ dmlZ ih\hjhlm 3 aZ aZdhghf ω =

= ω0 – $3 ^_ ω0 = 0,8 jZ^ k, : k-1 < fhf_gl qZkm W k dml 3 0 <bagZqblb dml ih\hjhlm 3 • dmlh\m r\b^d•klv ω \ fhf_gl

qZkm W k. jZ^ jZ^ k

2.14. I•^ qZk h[_jlZggy fZoh\bdZ ch]h dmlh\_ ijbkdhj_ggy af•gx\Zehky aZ aZdhghf ε : – <ω ^_ : jZ^ k2, < k-1 I_j_^ ]Zevfm\Zggyf

dmlh\Z r\b^d•klv fZoh\bdZ klZgh\beZ ω0 r\b^d•klv fZoh\bdZ ω q_j_a qZk W

\Zggy" jZ^ k

>BG:F1D: F:L?J1:EVGH2 LHQDB

L:IHKLMI:EVGH=H JMOM L1E

Hkgh\gL nhjfmeb

1fimevk l•eZ

→ →

P = mυ .

		JL\gyggy jmom lLeZ ^jm]bc aZdhg GvxlhgZ
					dP
			F = ma,		dt	= F ,
^_		n
	F	= ∑Fi	– j•\gh^•cgZ kbe yd• ^Lxlv gZ lLeh

i=1

12

KbeZ ly`•ggy

F = mg.

KbeZ l_jly dh\aZggy

F = kN.

^_ k – dh_n•p•}gl l_jly dh\aZggy N – kbeZ ghjfZevgh]h lbkdmAZdhg a[_j_`_ggy Lfimevkm

n

∑ Pi = const,

i=1

^_ n – d•evd•klv lLe sh \oh^ylv \ aZfdg_gm kbkl_fm

3.1.;jmkhd fZkhx P d] ly]gmlv aZ gbldm lZd sh \•g jmoZ}lvky a•

klZehx r\b^d•klx ih ]hjbahglZevg•c iehsbg• a dh_n•p•}glhf l_jly k = 0,14 <bagZqblb dml . f•` gbldhx • iehsbghx ijb

ydhfm gZly] gbldb [m^_ gZcf_grbf Qhfm ^hj•\gx} kbeZ gZly]m

Fmin? (80 G

3.2.IhobeZ iehsbgZ fh`_ af•gx\Zlb dml gZobem . ijb g_af•gg•c

^h\`bg• hkgh\b A €€ \_jogvh€ lhqdb \•evgh dh\aZ} g_\_ebd_ l•eh Dh_n•p•}gl l_jly l•eZ h[ ih\_jogx iehsbgb k = 0,14 Ijb ydhfm

dml• .0 gZobem iehsbgb ^h ]hjbahglm qZk dh\aZggy l•eZ [m^_ f•g•fZevgbf" (490)

3.3.FZl_j•ZevgZ lhqdZ fZkhx P d] jmoZ}lvky i•^ ^•}x kbeb a]•^gh

a j•\gyggyf x = A + Bt + Ct2 + Dt3 ^_ K	f k2, D = – f k3.
AgZclb agZq_ggy kbeb F \ fhf_gl qZkm t1	k; t2 k M ydbc

fhf_gl qZkm t3 kbeZ ^hj•\gx} gmex" G G k

3.4.L•eh fZkhx P d] jmoZ}lvky lZd sh aZe_`g•klv ch]h dhhj^b-

gZlb \•^ qZkm hibkm}lvky j•\gyggyf x = A cos ω t ^_ : f, ω jZ^ k AgZclb \_ebqbgm kbeb F, sh ^•} gZ l•eh \ fhf_gl qZkm t = 2 c. G

3.5.DmevdZ i•^\•r_gZ gZ g_jhaly`g•c gblp• ^h kl_e• \Z]hgZ ydbc

jmoZ}lvky ijyfhe•g•cgh \ ]hjbahglZevghfm gZijyfdm a]•^gh •a aZdhghf S = A + Ct2 ^_ : f, K f k2 GZ ydbc dml . gbldZ

\•^obey}lvky \•^ \_jlbdZe•" (80)

13

3.6.GZ ]hjbahglZevg•c iehsbg• a dh_n•p•}glhf l_jly k = 0,25 e_`blv

l•eh fZkhx P d] M fhf_gl qZkm W k ^h l•eZ ijbdeZeb ]h- jbahglZevgm kbem ydZ aZe_`blv \•^ qZkm yd F = Bt ^_ < H k. <bagZqblb reyo S ydbc ijhcreh l•eh aZ qZk W k ^•€ kbeb

f

3.7.GZ \•adm fZkhx m1 d] ydbc fh`_ \•evgh jmoZlbky \a^h\`

]hjbahglZevgbo j_chd e_`blv [jmkhd fZkhx m2 d] Dh_n•- p•}gl l_jly f•` [jmkdhf • \•adhf k = 0,25 ;jmkhd ly]gmlv a kbehx

F ydZ kijyfh\ZgZ iZjZe_evgh ^h j_chd • af•gx}lvky aZ aZdhghf

F = Bt ^_ < G c <bagZqblb ijbkdhj_ggy \•adZ .1 • [jmkdZ .2 \ fhf_gl qZkm W k. f k2 f k2)

3.8.GZ l•eh fZkhx P d] sh agZoh^blvky gZ ]eZ^d•c ]hjbahglZevg•c

ih\_jog• ^•} kbeZ ydZ ijhihjp•cgZ ^h qZkm F = Bt ^_ <	G c. Ydsh
t = 0, l•eh fZ} ihqZldh\m r\b^d•klv v0 f k	Ydbc reyo S
ijhc^_ l•eh aZ qZk W k? f

3.9.QZklbgdZ fZkhx P d] jmoZ}lvky ijyfhe•g•cgh • j•\ghf•jgh •a

r\b^d•klx v0 f k ih ]eZ^d•c ih\_jog• A ^_ydh]h qZkm gZ g_€ ihqbgZ} ^•ylb kbeZ F ydZ kijyfh\ZgZ \ [•d ijhlbe_`gbc ^h

gZijyfdm r\b^dhkl• Z fh^mev kbeb af•gx}lvky a qZkhf t aZ aZdhghf F = Bt ^_ < G c AgZclb fhf_gl qZkm t dheb

r\b^d•klv qZklbgdb ^hj•\gx} gmex lZ €€ r\b^d•klv v q_j_a

W k \•^ ihqZldm ^•€ kbeb k f k

3.10.L•eh fZkhx P d] \ fhf_gl qZkm t = 0 ihqZeh jmoZlbkv i•^ ^•}x

kbeb F = F0 sin ω t ^_ F0 G, ω jZ^ k <bagZqblb reyo S ydbc ijhcreh l•eh aZ qZk W k. f

3.11.GZ l•eh fZkhx P d] sh e_`blv gZ ]eZ^d•c ]hjbahglZevg•c iehsbg• \ fhf_gl qZkm t = 0 ihqZeZ ^•ylb kbeZ ydZ aZe_`blv \•^ qZkm aZ aZdhghf F = Bt ^_ < H k-1 GZijyfhd kbeb m\_kv qZk klZgh\blv dml . = 60º a ]hjbahglhf <bagZqblb r\b^d•klv v l•eZ \ fhf_gl \•^jb\Zggy \•^ iehsbgb • reyo S ydbc ijhcreh l•eh ^h pvh]h fhf_glm f k f

3.12. GZ l•eh fZkhx P d] ^•}	kbeZ ) G i•^ dmlhf . = 600 ^h
gZijyfdm jmom KbeZ l_jly	aZe_`blv \•^ r\b^dhkl• aZ aZdhghf

14

kgZjy^Z q_j_a qZk W k iZ^Z} gZ a_fex lhqgh i•^ f•kp_f \b[mom

<bagZqblb r\b^d•klv v2 ^jm]h€ qZklbgb kgZjy^Z \•^jZam i•key \b[mom f k

15

FT = F0 + Bv ^_ F0 G, < GÂk f <bagZqblb r\b^d•klv v •

ijbkdhj_ggy . l•eZ \ fhf_gl qZkm W k Z lZdh` r\b^d•klv v0 sh \klZgh\blvky ydsh \ fhf_gl qZkm t = 0 l•eh g_ jmoZehky f k

f k2 f k2)

3.13.:\lhfh[•ev fZkhx P d] a \bfdg_gbf ^\b]mghf fZxqb ihqZldh\m r\b^d•klv v0 f k amibgy}lvky i•^ ^•}x kbeb hihjm ydZ

	ijhihjp•cgZ ^h r\b^dhkl• Z\lhfh[•ey Fon = –rv				^_ U	GÂk f.
	AgZclb ]Zevf•\gbc reyo S Z\lhfh[•ey f
3.14.	L•eh	fZkhx P	d] jmoZ}lvky a• r\b^d•klx v0			f k •
	ihljZiey} m \¶yad_ k_j_^h\bs_			^_ gZ gvh]h	^•} kbeZ hihjm
	F = – Kv2 ^_ K GÂk2 f2 <bagZqblb ydhx [m^_ r\b^d•klv v
	jmom l•eZ \ k_j_^h\bs• q_j_a qZk W			k. f k
3.15.	Dmey jmoZxqbkv a• r\b^d•klx v1			f k ijh[b\Z} kl•gm lh\-
	sbghx • f • \be•lZ} a g_€ a• r\b^d•klx v2				f k KbeZ
	hihjm kl•gb ijhihjp•cgZ ^h dm[Z r\b^dhkl• dme• Fon = – Cv3.
	<bagZqblb qZk t jmom dme• \ kl•g• fk
3.16.	L•eh fZkhx m1		d] jmoZ}lvky gZamklj•q ^jm]hfm l•em fZkhx
	m2	d] <•^[mehky Z[khexlgh g_ijm`g_ a•ldg_ggy				pbo l•e
	R\b^dhkl• l•e [_aihk_j_^gvh i_j_^ m^Zjhf ^hj•\gx\Zeb v1					f k •
	v2	f k Dh_n•p•}gl l_jly k = 0,05 Kd•evdb qZkm t l•eZ [m^mlv
	jmoZlbkv i•key m^Zjm k
3.17.	Dmey fZkhx m1		d] jmoZ}lvky a• r\b^d•klx v1		f k • ijm`gh
	a•rlh\om}lvky a dme_x fZkhx m2			d] ydZ jmoZ}lvky a• r\b^d•klx
	v2	f k i•^ dmlhf . = 450 ^h ljZ}dlhj•€ i_jrh€ dme• <gZke•^hd
	m^Zjm ^jm]Z dmey \•^obebeZkv gZ dml 2 = 300 \•^ghkgh ihqZldh\h€
	ljZ}dlhj•€ i_jrh€ dme• Z €€ r\b^d•klv klZgh\beZ v2•				f k GZ ydbc
	dml 1 \•^obebeZkv i_jrZ dmey i•key a•ldg_ggy" (4106•
3.18.	KgZjy^ sh e_l•\ ]hjbahglZevgh a• r\b^d•klx v				f k gZ \bkhl•

Gf jhajb\Z}lvky gZ ^\• j•\g• qZklbgb H^gZ qZklbgZ

f k \•^ghkgh

3.19. M klhyq•c \h^• klh€lv g_jmohfh qh\_g fZkhx m1 d] Ex^bgZ fZkhx m2 d] ydZ agZoh^blvky m qh\g• i_j_creZ a ghkZ gZ dhjfm Qh\_g ijb pvhfm af•klb\ky gZ 6 f Hi•j \h^b ^m`_ fZebc YdZ ^h\`bgZ • qh\gZ" f

3.20.Ljb qh\gb dh`gbc fZkhx P d] jmoZxlvky h^bg aZ h^gbf a h^gZdh\hx r\b^d•klx v f k A k_j_^gvh]h qh\gZ h^ghqZkgh m

i_j_^g•c • aZ^g•c qh\gb db^Zxlv a• r\b^d•klx v1

qh\gZ ly]Zj• fZkhx m1 d] <bagZqblb r\b^dhkl• qh\g•\ i•key i_j_db^Zggy ly]Zj•\" f k f k f k

3.21. GZ i•^eha• klh€lv \•ahd m \b]ey^• ^h\]h€ ^hrdb fZkhx m1 d], ydZ fZ} e_]d• dhe_kZ GZ h^ghfm d•gp• ^hrdb klh€lv ex^bgZ fZkhx m2 d] Ex^bgZ ihqbgZ} jmoZlbkv \a^h\` \•adZ •a r\b^d•klx\•^ghkgh ^hrdb v2 f k A ydhx r\b^d•klx v1 \•^ghkgh i•^- eh]b [m^_ jmoZlbkv \•ahd" FZkhx dhe•k ag_olm\Zlb f k

JH;HL: 1 ?G?J=1Y Hkgh\gL nhjfmeb

Jh[hlZ ydZ \bdhgm}lvky klZehx kbehx

				r cosα,
	A = F	r = F
^_ α – dml fL` gZijyfZfb \_dlhjL\ kbeb F				L i_j_fLs_ggy r
Jh[hlZ afLggh€ kbeb
	A = ∫ F d ,
		L
^_ F• – ijh_dp•y kbeb F	gZ gZijyf _e_f_glZjgh]h i_j_f•s_ggy d
\a^h\` ljZ}dlhj•€ jmom L.
K_j_^gy ihlm`gLklv aZ Lgl_j\Ze qZkm t
	< N >=			A
				t

16

Fbll}\Z ihlm`gLklv

N = dAdt = Fυ cosα,

^_ α – dml f•` gZijyfZfb \_dlhj•\ kbeb F • r\b^dhkl• υ .

D•g_lbqgZ _g_j]•y fZl_j•Zevgh€ lhqdb Z[h ihklmiZevgh]h jmom l•eZ

Ek = m2v 2 .

L_hj_fZ ijh af•gm d•g_lbqgh€ _g_j]•€

A = Ek = Ek 2 − Ek1 .

Ihl_gp•ZevgZ _g_j]•y l•eZ \ ihe• a_fgh]h ly`•ggy

E p = mgh .

Ihl_gp•ZevgZ _g_j]•y l•eZ gZ yd_ ^•xlv kbeb ijm`ghkl•

= kx2 E p 2 ,

^_ x – af•s_ggy l•eZ \•^ghkgh iheh`_ggy j•\gh\Z]b

AZdhg a[_j_`_ggy f_oZg•qgh€ _g_j]•€ dhgk_j\Zlb\gh€ kbkl_fb

Ek + Ep = E = const .

Af•gZ f_oZg•qgh€ _g_j]•€ kbkl_fb

E = A ,

^_ :* – jh[hlZ g_dhgk_j\Zlb\gbo kbe yd• ^•xlv gZ l•eZ kbkl_fb

4.1.R\b^d•klv j_Zdlb\gh]h e•lZdZ gZ ^_yd•c ^•eygp• ljZ}dlhj•€ aZe_`blv \•^ ijhc^_gh]h reyom S aZ aZdhghf v = B + C S ^_ < f k, K k-1 FZkZ e•lZdZ m = 8 103 d] M fhf_gl qZkm

t1 k r\b^d•klv e•lZdZ v1		f k <bagZqblb jh[hlm ^\b]mg•\
aZ ijhf•`hd qZkm \•^ t1	k ^h t2 k. F>`
4.2. Dmey fZkhx m1 d]	sh jmoZ}lvky a• r\b^d•klx v1 f k,
gZa^h]Zgy} dmex fZkhx m2		d] sh jmoZ}lvky a• r\b^d•klx

v2 f k M^Zj dmev p_gljZevgbc <bagZqblb r\b^dhkl• dmev i•key ijm`gh]h ki•\m^Zjm f k f k

17

d] €€

dZ}lvky a g_jmohfhx dme_x fZkhx m2

4.3.Dmey fZkhx m1 = d] sh jmoZ}lvky a ^_ydhx r\b^d•klx v1 klb-

d] Ki•\m^Zj dmev Z[kh-

exlgh ijm`gbc • p_gljZevgbc Ydm qZklbgm w k\h}€ d•g_lbqgh€ _g_j]•€ i_jrZ dmey i_j_^ZeZ ^jm]•c" (0,75)

4.4. KZgdb yd• jmoZebky ih evh^m a• r\b^d•klx v f k \b€`^`Zxlv

gZ ZknZevl >h\`bgZ iheha•\ kZghd / f dh_n•p•}gl l_jly h[ ZknZevl k = 0,5 <\Z`Zlb sh fZkZ jhaih^•e_gZ ih ^h\`bg• kZghd

j•\ghf•jgh Ydbc reyo S ijhc^_ i_j_^g•c d•g_pv kZghd ih ZknZevlm ^h ih\gh€ amibgdb" f

4.5.Ih ihobe•c iehsbg• a dmlhf gZobem . = 300 a¶€`^`Z} eb`gbd

fZkhx m1 d] Ijh€oZ\rb \•^klZgv • f \•^ \_jrbgb \•g klj•ey} \]hjm kb]gZevghx jZd_lhx FZkZ jZd_lb m2

ihqZldh\Z r\b^d•klv v2 f k L_jly g_ \jZoh\m\Zlb <bagZqblb r\b^d•klv v1 eb`gbdZ i•key ihklj•em f k

4.6.A \_jrbgb •^_Zevgh ]eZ^dh€ kn_jb a•kdh\am} g_\_ebdbc \ZglZ` JZ^•mk kn_jb 5 f GZ yd•c \bkhl• h \•^ gbam kn_jb \ZglZ` a•j\_lvky a g_€" f

4.7.Ihl_gp•ZevgZ _g_j]•y qZklbgdb \ p_gljZevghfm kbeh\hfm ihe• aZ^ZgZ yd nmgdp•y \•^klZg• r \•^ p_gljZ ihey ^h lhqdb ^_ i_j_[m- \ZeZ qZklbgdZ

U (r) = Ar + rB2 ,

^_ : 10-4 >` f, < 10-6 >` f2.

<bagZqblb aZ ydbo agZq_gv r ihl_gp•ZevgZ _g_j]•y • kbeZ sh ^•} gZ qZklbgdm fZxlv _dklj_fZevg• agZq_ggy • agZclb p• agZq_ggy

kf kf – f>` – G

4.8.:\lhfh[•ev jmoZ}lvky i•^ ^•}x kbeb ly]b F ydZ af•gx}lvky aZe_`gh \•^ ijhc^_gh]h reyom aZ aZdhghf F = B + C S + D S2 ^_

<G, K G f, D = 3 H f2 <bagZqblb jh[hlm kbeb gZ

^•eygp• reyom \•^ S1 f ^h S2 f. >`

4.9.GZ fhlhjgbc qh\_g ydbc jmoZ}lvky gZ i•\g•q ^•} kbeZ \•ljm F =

G. Dml f•` gZijyfhf ^•€ kbeb F • gZijyfhf jmom qh\gZ af•-

18

kf.

	gx}lvky aZ aZdhghf . = B S ^_ <			10-2 jZ^ f GZijyf \•ljm af•-
	gb\ky a i•\^_ggh]h gZ ko•^gbc AgZclb jh[hlm \•ljm >`
4.10.	KbeZ ly]b Z\lhfh[•ey af•gx}lvky aZe_`gh \•^ reyom aZ aZdhghf
	F = B + C S ^_ <	G, K G f H[qbkeblb jh[hlm : kbeb
	gZ ^•eygp• reyom \•^ S1		f ^h S2	f. >`
4.11.	Ijbkdhj_ggy qh\gZ	gZ	i•^\h^gbo	djbeZo af•gx}lvky aZe_`gh \•^

reyom aZ aZdhghf a = B + C S + D S2 ^_ < f k2, K 10-4 k2, D = 3 10-6 f c2) FZkZ qh\gZ m = 5 103 d] <bagZqblb jh[hlm :

i_j_f•s_ggy qh\gZ gZ ^•eygp• reyom \•^ S1 f ^h S2 = 103 f.

F>`

4.12.<•ljbevgbd fZkhx P d] jmoZ}lvky i•^ ^•}x klZeh€ kbeb ijyfhe•g•cgh ijbqhfm aZe_`g•klv ijhc^_gh]h reyom S \•^ qZkm t

\bagZqZ}lvky \bjZahf S = B t + C t2 ^_ < f k, K f k2. AgZclb jh[hlm : kbeb \•ljm aZ ijhf•`hd qZkm \•^ t1 = 0 ^h t2 k.

d>`

4.13.R\b^d•klv ih€a^Z fZkZ ydh]h m = 105 d] af•gx}lvky aZ aZdhghf

v = B + D t2 ^_ < f k, ' f k3 <bagZqblb jh[hlm kbeb ly]b aZ qZk \•^ t1 k ^h t2 = 30 c. F>`

4.14. <k_j_^bg• ljm[b [•ey €€ djZx agZoh^blvky dhjhd ^h\`bghx •0 FZdkbfZevgZ kbeZ l_jly f•` dhjdhf • ljm[hx FT = G Kl•gdb ljm[b klbkdZxlv dhjhd ih \k•c ^h\`bg• j•\ghf•jgh Ydm jh[hlm ihlj•[gh \bdhgZlb sh[ \bly]gmlb dhjhd a ljm[b" >`

4.15.	<•l_j ydbc ^f_ a• r\b^d•klx v0 f k ^•} gZ \•ljbeh iehs_x
	6 f2 a• kbehx
	F = ASρ (v0 −v )2	,
	2
	^_ A – ^_ydbc [_ajhaf•jgbc dh_n•p•}gl :		, ! – ]mklbgZ
	ih\•ljy ! d] f3), v – r\b^d•klv km^gZ <bagZqblb fZdkb-
	fZevgm fbll}\m ihlm`g•klv N \•ljm d<l
4.16.	FZl_j•ZevgZ lhqdZ fZkhx P d] jmoZeZky i•^ ^•}x ^_ydh€ kbeb
	a]•^gh a j•\gyggyf S = A + Bt + Ct2 + Dt3, ^_ :		f, < – f k,

Kf k2, D = – f k3 <bagZqblb ihlm`g•klv N ydZ aZljZqZ}lv-

ky gZ jmo lhqdb \ fhf_gl qZkm W k <l

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4.17. QZklbgdZ fZkhx P ] jmoZ}lvky \ ^\h\bf•jghfm ihe• \ ydhfm €€

ihl_gp•ZevgZ _g_j]•y Ej	= Bxy ^_ < f>` f2 M lhqp• a dhhj-
^bgZlZfb x1 f, y1	f qZklbgdZ fZeZ r\b^d•klv v1 f k Z \
lhqp• a dhhj^bgZlZfb x2	f, y2 f r\b^d•klv v2 f k <bagZ-

qblb jh[hlm klhjhgg•o kbe gZ reyom f•` pbfb lhqdZfb f>`

>BG:F1D: H;?JL:EVGH=H JMOM

L<?J>H=H L1E:

Hkgh\gL nhjfmeb

Fhf_gl kbeb F \L^ghkgh aZ^Zgh€ ]h lhqdb p_gljZ 0

= [

M rF

^_ r – jZ^Lmk-\_dlhj ijh\_^_gbc a lhqdb p_gljZ 0 ^h lhqdb ijbdeZ-

^Zggy kbeb F .

Fhf_gl kbeb \•^ghkgh hk• z

M z = ±F z ,

^_ F – \_ebqbgZ kbeb kdeZ^h\h€ kbeb ydZ ^•} \ i_ji_g^bdmeyjg•c ^h hk•

ziehsbg• •z – ie_q_ kbeb \•^ghkgh hk• z.

Fhf_gl Lfimevkm fZl_jLZevgh€ lhqdb \L^ghkgh aZ^Zgh€ ]h lhqdb

p_gljZ

L= r P = [r mυ].

Fhf_gl Lfimevkm fZl_jLZevgh€ lhqdb \L^ghkgh aZ^Zgh€ hk• z→

Lz = mv z

^_ •z – ie_q_ Lfimevkm fZl_jLZevgh€ lhqdb \•^ghkgh aZ^Zgh€ hk• z.

J•\gyggy fhf_gl•\
d L		d L	= M z .
d t	= M ;	z
		d t

Fhf_gl •g_jp•€ fZl_jLZevgh€ lhqdb \L^ghkgh hkL

J = mr 2 ,

^_ r – \L^klZgv lhqdb \L^ hkL

20