ІКТА / КБ-12 / Фізика / Збірник задач з фізики Лопатинський І.Є

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D1G?F:LBD: F:L?J1:EVGH2 LHQDB

L:IHKLMI:EVGH=H JMOM L1E

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DLg_fZlbqgL jL\gyggy jmom \ dhhj^bgZlgLc nhjfL x = x(t); y = y(t); z = z(t).

DLg_fZlbqg_ jL\gyggy \ ijbjh^g•c nhjfL s = s(t),

^_ s – reyoh\Z dhhj^bgZlZ

Ijh_dpL€ r\b^dhklL gZ dhhj^bgZlgL hkL

<_ebqbgZ fh^mev fbll}\h€ r\b^dhkl•

υ= υx2 +υy2 +υz2 .

J•\gyggy nhjfmeZ reyom

S = S(t).

Fbll}\Z r\b^dLklv

υ= ddSt = S.

Reyo ijhc^_gbc l•ehf

t2

S = ∫υ(t)dt.

t1

K_j_^gy r\b^d•klv jmom

<υ >= St .

Ijh_dpL€ ijbkdhj_ggy gZ hkL dhhj^bgZl

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az = ddυtz =υz = z.

<_ebqbgZ fh^mev ijbkdhj_ggy

a= ax2 + ay2 + az2 .

<_ebqbgZ fh^mev lZg]_gpLZevgh€ kdeZ^h\h€ ijbkdhj_ggy

aτ = ddυt

<_ebqbgZ fh^mev ghjfZevgh€ kdeZ^h\h€ ijbkdhj_ggy

D3 = υ5

^_ R – jZ^•mk djb\bgb ljZ}dlhj•€

1 <_ebqbgZ fh^mev ih\gh]h ijbkdhj_ggy a = aτ2 + an2 .

K_j_^g} ijbkdhj_ggy

< a >= υ2 −υ1 t2 −t1

DLg_fZlbqg_ jL\gyggy jL\ghfLjgh]h ijyfheLgLcgh]h jmom

x= xo +υx t.

DLg_fZlbqg_ jL\gyggy jL\ghafLggh]h ijyfheLgLcgh]h jmom

x= xo +υoxt + ax2t2 .

1.1.>\Z Z\lhfh[•e• jmoZxlvky ih ^\ho ijyfhe•g•cgbo • \aZ}fgh i_ji_g^bdmeyjgbo ^hjh]Zo m gZijyfdZo ^h i_j_oj_kly a• klZebfb

df ]h^ • v2 df ]h^ I_j_^ ihqZldhf jmom i_jrbc Z\lhfh[•ev i_j_[m\Z\ gZ \•^klZg• S1 = df \•^ i_j_oj_kly ^jm]bc gZ \•^klZg• S2 df Q_j_a ydbc qZk i•key ihqZldm jmom \•^klZgv f•` Z\lhfh[•eyfb [m^_ f•g•-

fZevghx" ]h^)

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1.2.Jmob ^\ho fZl_j•Zevgbo lhqhd aZ^Zxlvky j•\gyggyfb x1 = A1 +

+ B1t + C1t2, x2 = A2 + B2t + C2t2 ^_ A1 f, B1 f k,

C1 = – f k2, A2 f, B2 f k, C2 f k2 < ydbc fhf_gl qZkm t r\b^dhkl• pbo lhqhd [m^mlv h^gZdh\bfb" <bagZqblb

r\b^dhkl• v1 • v2 lZ ijbkdhj_ggy .1 • .2 lhqhd m p_c fhf_gl

k f k f k2 f k2)

1.3.>\• fZl_j•Zevg• lhqdb jmoZxlvky a]•^gh a j•\gyggyfb x1 = A1 +

+ C1t2 + D1t3, x2 = B2t + C2t2 + D2t3 ^_ A1 f, C1 f k2,

D1 = – f k3, B2 f k, C2 2, D2 f k3 < ydbc fhf_gl qZkm ijbkdhj_ggy pbo lhqhd [m^mlv h^gZdh\bfb" <bagZqblb r\b^dhkl• lhqhd m p_c fhf_gl k f k f k

1.4.H^ghqZkgh a h^gh]h • lh]h kZfh]h imgdlm \ h^ghfm gZijyfdm

f

1.7.AZe_`g•klv r\b^dhkl• l•eZ \•^ qZkm aZ^Z}lvky j•\gyggyf v = A +

f k3 Ydbc reyo

k ^h t2 k" YdZ k_j_^gy r\b^d•klv <v> l•eZ • k_j_^g} ijbkdhj_ggy . ! aZ p_c ijhf•`hd qZkm" f f k f k2)

1.8.AZe_`g•klv r\b^dhkl• l•eZ \•^ qZkm hibkm} j•\gyggy v = A + Bt ^_ : f k, < f k2 Ydbc reyo ijhoh^blv l•eh aZ ijhf•`hd qZkm

\•^ t1 k ^h t2 k" <bagZqblb k_j_^gx r\b^d•klv <v> l•eZ aZ p_c ijhf•`hd qZkm f f k

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1.9.F¶yq dbgmeb a i•^\bs_ggy \ ]hjbahglZevghfm gZijyfdm a•

I•key ihqZldm jmom \hgZ ijhoh^blv reyo ydbc ^hj•\gx} 0,2

^h\`bgb dheZ <bagZqblb ih\g_ ijbkdhj_ggy . lhqdb m p_c fhf_gl qZkm f k2)

1.11.GhjfZevg_ ijbkdhj_ggy lhqdb sh jmoZ}lvky ih dhem jZ^•mkhf

5 f, af•gx}lvky aZ aZdhghf .n = A + Bt + Ct2 ^_ : f k2, < f k3, K f k4 AgZclb lZg]_gp•Zevg_ ijbkdhj_ggy .2 lhqdb

reyo ydbc \hgZ ijhcreZ aZ qZk t1 k i•key ihqZldm jmom ih\g_ ijbkdhj_ggy . m fhf_gl qZkm t2

1.12. Jmo fZl_j•Zevgh€ lhqdb \ iehsbg• xy hibkm}lvky j•\gyggyf x = At, y = At + Bt2 ^_ : f k, < – f k2 HljbfZlb j•\gyggy

ljZ}dlhj•€ fZl_j•Zevgh€ lhqdb y(x) r\b^d•klv v lhqdb \ fhf_gl

qZkm t1 k ijbkdhj_ggy . lhqdb fhf_gl qZkm t2 m ydbc dml f•` r\b^d•klx • ijbkdhj_ggyf . Œ . (y = 0,5 + 4x2 f k

f k2 k

1.13.J•\gyggy jmom fZl_j•Zevgh€ lhqdb fZxlv \b]ey^ x = A1 + B1t,

y = B2t + C2t2, z = 0 ^_ :1 = – f, <1 f k, <2 f k,

K2 = – f k2 <bagZqblb r\b^d•klv v ijbkdhj_ggy . lZ dml f•` \_dlhjZfb pbo \_ebqbg m fhf_gl qZkm dheb dhhj^bgZlZ y gZ[m\Z} _dklj_fZevg_ agZq_ggy f k f k2; 900)

1.14. J•\gyggy jmom fZl_j•Zevgh€ lhqdb fZxlv \b]ey^ x = B1t + D1t3, y = B2t + C2t2, z = 0 ^_ <1 f k, D1 = – f k3, B2 f k,

C2 = – f k2 <bagZqblb lZg]_gp•Zevg_ .2 , ghjfZevg_ .n ijbkdh- j_ggy • jZ^•mk djb\bgb R ljZ}dlhj•€ \ fhf_gl qZkm t1 dheb dhhj-

^bgZlZ y gZ[m\Z} _dklj_fZevg_ agZq_ggy • \ fhf_gl t2 dheb lZd_ agZq_ggy gZ[m\Z} dhhj^bgZlZ x. ( f k2 f k2, 1 f f k2;

f k2 f)

1.15. Jmo fZl_j•Zevgh€ lhqdb ih dhem jZ^•mkhf 5 f hibkm} j•\gyggy S = A + Ct2 ^_ : f, K – f k2 <bagZqblb fhf_gl qZkm t dheb

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ghjfZevg_ ijbkdhj_ggy lhqdb ^hj•\gx} .n f k2 AgZclb

r\b^d•klv v lZg]_gp•Zevg_ .2 • ih\g_ . ijbkdhj_ggy lhqdb \ p_c fhf_gl qZkm k f k f k2 f k2)

1.16. AZe_`g•klv reyom \•^ qZkm ^ey lhqdb sh jmoZ}lvky ih dhem jZ^•mkhf 5 f hibkm} j•\gyggy S = Dt3 ^_ ' f k3. <bagZqblb fhf_gl qZkm t dheb e•g•cgZ r\b^d•klv lhqdb v f k.

AgZclb ghjfZevg_ .n lZg]_gp•Zevg_ .2 • ih\g_ . ijbkdhj_ggy lhqdb \ p_c fhf_gl qZkm k f k2 f k2 f k2)

1.17.FZl_j•ZevgZ lhqdZ ihqbgZ} jmoZlbky aZ ]h^bggbdh\hx klj•edhx ih dhem jZ^•mkhf 5 f •a klZebf lZg]_gp•Zevgbf ijbkdh- j_ggyf .2 = 0 f k2 Q_j_a ydbc qZk t \_dlhj ijbkdhj_ggy . ml\hjblv a \_dlhjhf r\b^dhkl• v dml . = 600" Ydbc reyo ijhc^_ aZ p_c qZk jmohfZ lhqdZ" k f

1.18. FZl_j•ZevgZ lhqdZ jmoZ}lvky ih dhem jZ^•mkhf 5 f M fhf_gl qZkm dheb \_dlhjb ih\gh]h . • ghjfZevgh]h .n ijbkdhj_gv ml\h-

f k f k

2.D1G?F:LBD: H;?JL:EVGH=H JMOM L<?J>H=H L1E:

Hkgh\gL nhjfmeb

DLg_fZlbqg_ jL\gyggy h[_jlZevgh]h jmom

ϕ = ϕ(t).

Fbll}\Z dmlh\Z r\b^dLklv

ω = ddϕt .

K_j_^gy dmlh\Z r\b^dLklv

<ω >= ϕ2 −ϕ1 . t2 −t1

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<_ebqbgZ fh^mev dmlh\h]h ijbkdhj_ggy

ε= ddωt = ω = ϕ .

DLg_fZlbqg_ jL\gyggy jL\ghfLjgh]h h[_jlZevgh]h jmom

ϕ= ϕ0 + ωt.

I_jLh^ h[_jlZggy

T = 2ωπ .

QZklhlZ h[_jlZggy

n = T1 , n = 2ωπ .

ELgLcgZ r\b^dLklv

υ= ωR.

LZg]_gpLZevg_ ijbkdhj_ggy

aτ = εR.

GhjfZevg_ ijbkdhj_ggy

an = ω2 R.

qZk t mijh^h\` ydh]h a^•ckgx\Zehkv ]Zevfm\Zggy jZ^ k2;

k

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