Motion in One Dimension

7

Dntfnm fm Nmk Hfdkmsfnm Fmtrnhugtnry Kxkrgfsk 7.0

0. Wuppnsk

` p`rtfgok fs dnvfmj wftb gnmst`mt vkongfty v (ònmj tbk `xfs na x). x).

4 vt0 Hfspo`gkdkmt na p`rtfgok fm tfdk t< 4 vt< ∸ Hfspo`gkdkmt na tbk p`rtfgok fm tbk tfdk fmtkrvò ∁t ( 4 t< ∜ t0) 4 vt< ∜ vt0 4 v ( t< ∜ t0) vkongfty fm tbk tfdk tfdk fmtkrvò ∸ @vkr`jk vkongfty v ( t< ∜ t0) ∁t ‼ 4 ( t< ∜ t0) 4 v Hfspo`gkdkmt na p`rtfgok fm tfdk t0

Mnw, `s tbk p`rtfgok fs dnvfmj wftb gnmst`mt vkongfty (f.k (f.k., ., wftb gnmst`mt spkkh fm ` jfvkm hfrkgtfnm) fts vkongfty `mh spkkh `t `my fmst`mt wfoo nlvfnusoy lk v.

4 <.5 gd Ú υ r`h s ∜0 75 υ 4 gds ∜0 0;

@s tbk tfp wnuoh lk dnvfmj wftb gnmst`mt spkkh. υ gds ∜0 @vkr`jk spkkh 4 0; Fm 0; s tbk skgnmh b`mh wnuoh rnt`tk tbrnujb 65µ f.k., f.k., tbk hfspo`gkdkmt na fts tfp wfoo lk r <.

∸ Dnhuous na `vkr`jk vkongfty na tbk tfp na skgnmh b`mh fm 0; s.

4 4

r < 0; < < 0;

gds ∜0

8. (`) Vks. Ly gb`mjfmj hfrkgtfnm na dntfnm,

@ms. Xruk. <. @s tbk stnmk wnuoh lk arkk tn `ggkokr`tfnm

umhkr k`rtb‖s jr`vfty ft `ggkokr`tfnm wfoo lk j. j. f.k., 15s tn 7. @ skgnmh b`mh tìks 0 dfm f.k., gndpoktk nmk rnt`tfnm (f.k., (f.k., rnt`tfnm rnt`tfnm ly `m `mjok na <υ r`h).

∸ @mjuo`r spkkh na skgnmh b`mh <υ r`h 4 15 s

r`h s ∜0 4 υ r` 75

Ofmk`r spkkh na fts tfp 4 r`hfus Ú `mjuo`r spkkh

tbkrk wfoo lk gb`mjk fm vkongfty `mh sn `ggkokr`tfnm. (l) (f) Mn. Fm gurvkh p`tb tbkrk wfoo òw`ys lk `ggkokr`tfnm. (@s kxpo`fmkh fm tbk prkvfnus `mswkr mn. 7) (ff) Vks. Fm prnckgtfok dntfnm dntfnm tbk p`tb na tbk p`rtfgok fs ` gurvkh nmk wbfok `ggkokr`tfnm na tbk p`rtfgok rkd`fms gnmst`mt. (fff) Vks. Fm gurvkh gurvkh p`tb p`tb tbk `ggkokr`tfnm `ggkokr`tfnm wfoo òw`ys lk tbkrk. Kvkm fa tbk p`tb fs gfrguo`r wftb gnmst`mt spkkh tbk hfrkgtfnm na tbk `ggkokr`tfnm na tbk p`rtfgok wnuoh kvkry tfdk lk gb`mjfmj.

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Dntfnm fm Nmk Hfdkmsfnm |

;. (`) Xfdk spkkh

4 Gfrgudakrkmgk 4

4 8
Wpkkh <υ Ú 8 gd

4 2 υ 4 <;.07

0gd/s

4

8r <

4<

 <υr     spkkh

4<

<

υ

v

s

(l) @s p`rtfgok p`rtfgok fs dnvfmj wftb gnmst`mt spkkh na 0 gd/s, fts `vkr`jk spkkh fm `my tfdk fmtkrvò wfoo lk 0 gd/s. r < | @vkr`jk vkongfty vkongfty|4 X / 8

4

<

υ

0;

spkkh

gds ∜0

4 5.6 gds ∜0 v | @vkr`jk `ggkokr`t `ggkokr`tfnm fnm|4

< X / 8

< < v< υ r

4<

< (0)< υ 8

.<7 gd s ∜< 4 5.<7 1. Hfst`mgk 4 Wpkkh Ú tfdk H0 4 v0t0 H< 4 v< t< H + H< v0t0 + v< t< @vkr`jk spkkh 4 0 4 t0 + t< t0 + t< ( 8 Ú <) + (1 Ú 7) 4 <+7

4 ;.
(wbkrk v 4 spkkh)

Fmtrnhugtnry Kxkrgfsk 7.< 0. @ggkokr`tfnm (huk tn jr`vfty).

v 4 t 7 / 8 hs 4 t7 / 8 ht

8.

0 ` fs pbysfgòoy gnrrkgt `s ft < jfvks tbk hfspo`gkdkmt na tbk p`rtfgok fm ttb skgnmh (nr `my tfdk umft).

<. st 4 u + `t ∜

∸

s 4

∩ t

s 4

8

st 4 Hfspo`gkdkmt fm t skgnmhs skgnmhss ∜ hfspo`gkdkmt fm ( t ∜ 0) skgnmh 4 ut + 0 `t<  ∜ u ( t ∜ 0) + 0 `( t ∜ 0)<  < <     st 4 st 4 u + `t ‗ `t ‗

<

`

Hfspo`gkdkmt Qkongfty

@ggkokr`tfnm

Xbkrkanrk, tbk jfvkm hfdkmsfnmòoy fmgnrrkgt.

kqu`tfnm

7 / 8

(jfvkm) …(f) 7

ht 4

+0 t8 7 +0 8

+g

+g : f.k., f.k., s ∔ t : / 8 Hfaakrkmtf`tfmj Kq. (f) w.r.t. tfdk t, 7 h< s 7 8 ∜ 0 4 t ht< 8 ` ∔ t∜0 / 8 ⇜ nr

t : / 8

s) na tbk p`rtfgok ;. Hfspo`gkdkmt ( s) fs

s 4 ( 85 Ú 1) +

0 (∜ 05) 1< <















01 | Dkgb`mfgs-0

4 85 4 8 s ? 1 s

∸

05

Ft fdpofks tb`t p`rtfgok b`s gndk l`gi àtkr `tt`fmfmj d`xfdud bkfjbt (b (b) jfvkm ly u< b4 < j ( 85)< 4 25 d 25 d < Ú 05

4 ∸

∸ nr

hx

1.

0<; 4 5 ⋋ t +

6.

t 4 <; s <; s 0<; d @vkr`jk vkongfty vkongfty 4 ;s

(hnwmw`rhs) (hnwmw`rhs ) …(f

) hv ht

∸

`

4

@t

t 4 < s

4 ; ∜
` 4 ; ∜ < Ú <

4 0 d/s<

4 ( 85 ∜ 05t) ht x 4 ∩ ( 85 ∜ 05t) ht x 4 85t ∜ ;t< + g

nr

(hnwmw`rhs)

4 <; d/s v 4 05 + ;t ∜ t<

ht

nr

0 < jt <

⇜

05.

H 4 25 + (25 ∜ 15)

4 055 d 055 d v 4 85 ∜ 05t hx 4 85 ∜ 05t

@ms. @ms. Xruk.

Arnd Kq. (f), hx 4 05 + ;t ∜ t< ht

∩ (05 +;;t t ∜ tt ) ht

@s `t t 4 5 tbk vòuk na x fs x fs zkrn.

∸

x4

g 4 5

nr

x 4 05t +

@s, `t

t 4 5 tbk vòuk na x fs x fs zkrn

∸

x 4 85t ∜ ;t

<

Anr x tn x tn lk 15 d.

∜ 2 t + 0< 4 5 ∸ t 4 < s nr 1 s Hfspo Hfspo`gk `gkdk dkmt mt fm tfdk tfdk t t vkongfty 4 :. @vkr`jk vkongfty t<

∸

x 4 05t +

Xbus, `t

t 4 7 s

0 < `t < t 0 `t <

ut +

4u+ ∸

↜ 00. u ↜ `

4 v0 + `t `t 4 v< ∜ v0

@vkr`jk vkongfty vkongfty 4

4

Hfspo Hfspo`gk `gkdk dkmt mt fm tfdk tfdk t t

; <

<

t

0 < `t < t 0

∜

7

+g

t7 7

; < (7) <

∜7

4 < ]f d/s 4 (< gns 15µ ]f + < sfm 15µ c]) d/s< 4 (0 ]f +

]

7 c) d/s< y

t v0t +

∜

7

4 75 + <<.; ∜ 6 4 87.; d

v<

2.

<

x 4 (05 Ú 7) +

t

4

<

g 4 5

15 4 85t ∜ ;t< nr

<

` 4 < d/s < 15µ

7

7
















↜ |v| 4 8< + 0< 4 < : d/s ↜ ↜ 0↜ s 4 u t + ` t< < 4 (< ]f) < + 0 (0 ]f + 7 ]c) << <

4 8 ]f + < ]f + < 4 1 ]f + < ↜ |s |4

∸ 0<. U`rt F ↜

U`rt FF Vks. @s kxpo`fmkh lkonw. lkonw. ↜ ] ] v 4 < f +
tbk p`rtfgok fs < f d/s< `mh tbk `ggkokr`tfnm fs < c d/s c d/s< ↜ ∸ s (`t t 4 0 s) 4 (< ]f Ú 0) + 0 (< ]c) 0< <

]

7 c

4 (< ]f + ]c) d

]

7 c 07. x

71 + 0<

4 8 7 d ↜ ] ] v 4 (< f +
…(f)

<

∸

y 4   x    <

nr,

x<

4 8y

x 4
∸

hy

∸

ht ↜ v

Xbus,

∩

4
↜

4 < f.k., v x 4 < ]f

y 4 t<

Arnd Kq. (f), ↜ hs 4 (< ]f +
4
(Xbk `lnvk fs tbk kqu`tfnm tn tr`ckgtnry)

hv 4 < c] ht ↜ ] ` 4 < c d/s<

↜ s

↜

4
↜

4 v x + v y 4 (< ]f +
c+ g

< ]

Xìfmj fmftfò hfspo`gkdkmt tn lk zkrn. ↜ ] ] s (`t t 4 0 s) 4 (< f + c) d

↜ `

4

↜ hv ht

4 < c] d/s<

Fmtrnhugtnry Kxkrgfsk 7.7 0. @t t

4 t0 v

s

ΰ

t <

μ

μ t

t 0 5

0:

β

t 0

t <

v 4 t`m β @s β ? 65µ vt fs + fvk.

ρ

t

@ggkokr`tfnm @ggkokr`tfnm `t t 4 t0 3 `t0

4 t`m ΰ

@s ΰ ? 65µ ` t fs + fvk gnmst`mt.















02 | Dkgb`mfgs-0 0 dj< t< < < f.k., f.k., IK ∔ t . Zbfok jnfmj up tbk vkongfty wfoo jkt ∜ fvk lut tbk IK wfoo rkd`fm. IK wfoo rkhugk tn zkrn `t tfdk < t wbkm tbk p`rtfgok rk`gbks fts fmftfò pnsftfnm.

Gnrrkspnmhfmj vkongfty-tfdk wfoo lk

wnuoh lk jt. jt. IK na p`rtfgok wfoo lk

Qkongfty (d/s)

8

2

t (s)

IK

8.

t 4

IK 4

0 dj< t< <

tfdk

b ( t ∜ <)

4 t`m β 4

⇜

< (< ∜ 0) b 4 < ( t ∜ <)

` (d/s<)

4

0
<

β

4 djb

0

7. Wpkkh na lòo (cust lkanrk dìfmj afrst

<

β

( t ‗ < ) t

gnoofsfnm wftb aonnr)

4 < jb 4 4 85 d/s 85 d/s

Xfdk tìkm tn rk`gb jrnumh < Ú 25 4
48s

Wpkkh na lòo (cust àtkr afrst gnoofsfnm wftb aonnr) 4 85 4 <5 d/s < Xfdk tn `tt`fm d`xfdud bkfjbt ∜ <5 4 < s t 4 ∜05

∸ Xfdk anr tbk rkturm cnurmky tn aonnr 4 < s. Wpkkh (d/s)

8

b

< Ú 05 Ú 25

2

t (s)

U`rtfgok wfoo `tt`fm fts fmftfò vkongfty f.k., mkt fmgrk`sk fm vkongfty na tbk p`rtfgok wfoo lk zkrn wbkm, `rk` umhkr `-t jr`pb 4 5 (0 + <) Ú < ( ∜ b) ( t ∜ <) <

+

<

45

∜ ( t ∜ <)< 4 5 nr ( t ∜ <)< 4 7 nr t ∜ < 4 ° 7 nr t 4 < ° 7 @ms . @ms . @t tfdk t 4 < + 7 s (t 4 < ∜ 7 mnt pnssflok). nr

7
















s 4 (v @

nr

sfm 75µ 0;5 t 0 755

∜ vL ) t

s

β t

7. Fm afjurk, u 4 spkkh na ln`t

v 4 spkkh na rfvkr aonw L

G @gtuò p`tb na ln`t

Ln`t s`fofmj hfrkgtfnm

nr

u h 4 855 d

⇜

@ v

  + v< 

05 d/s

4

05 d/s

Ú ( 855 d )

4 25 d 8. Okt G lk tbk pnfmt ònmj wbfgb pfont

sbnuoh bk`h tbk po`mk.

` L 4 < d/s<

v L 4 0 d/s

Hrfatfmj huk tn wfmh

∜ 0 4 < d/s Fmftfò hfspo`gkdkmt na @ w.r.t. @ w.r.t. L L 4 ∜ 05 d

@t tfdk rko`tfvk hfspo`gkdkmt hfspo`gkdkmt na @ na @ w.r.t. w.r.t. L L 0 s 4 ∜ 05 + < t + ( ∜ 0) t< < s4

∜ 05 +
Anr s tn s tn lk dfmfdud hs 45 ht < ∜ (5.; Ú
f.k., f.k., Wpkkh 4 0;5 d/s

∜ 0 d/s<

Qkongfty na @ w.r.t. @ w.r.t. L L 4 7

nr µ 5 7 β

L

` @ 4 0 d/s< ,

nr

L d i 5 5 ;

@

@ggkokr`tfnm na @ w.r.t. @ w.r.t. L L 4 0 ∜ < 4

u < d/s

Mnrtb

05 d

;.

v @ 4 7 d/s,

4 85 s v LG 4 @L

0

<626 s 4 <626 s ;5 dfm 4 ;5 dfm

@G u<

∜ 75µ + β) ;55 Ú 057

;555 0 <<8 4 + 7⋋ 0 7t < 0; < 0; ;555 4 5.;;:: 7 t ;555 t 4 7 Ú 5.;;::

Xfdk tn grnss rfvkr

 4 @L  4 LG 4 u   v 855 d 4

sfm (025µ

<5t 4 sfm β <5 β 4 sfm ∜0   0    0; 0; sfm (75µ + β) 0 β Mnw 4 755 t ;55 Ú 057 <<8 ;555 nr 4 sfm 75µ gns β + gns 75µ sfm β 7t

t`m β 4 (v (v ‗ v L) @ ‗v

5

4 sfm β 4

06

∸

sdfm

4 ∜ 05 + (< Ú <) ∜ 5.; Ú (<)< 4 ∜ 05 + 8 ∜ <

















<5 | Dkgb`mfgs-0

@FKKK Gnrmkr Wulckgtfvk Pukstfnms (Okvko 0) 0. (`) H 4 v0t0

+ v< t< 4  15 id Ú 0 b  + +  25 id Ú 0 b   b   b <  4 055 id

(l) @vkr`jk spkkh 055 055 id

4

0.; b

4 11.1: id/b. 11.1: id/b.

<. (`) Hfspo`gkdkmt fm afrst twn skgnmhs

4 ( 8) (<) + 0 (1) << <

4 <5 d ∸

@vkr`jk vkongfty 4

<5 d
4 05 d/s

(l) Hfspo`gkdkmt fm afrst anur skgnmhs 4 ( 8) ( 8) + 0 (1) 8< 4 18 d <

∸ Hfspo`gkdkmt fm tbk tfdk fmtkrvò t 4 < s tn t 4 8 s 4 18 ∜ <5 4 88 d 88 d 88 d ∸ @vkr`jk vkongfty 4 4 << d/s.
7. Okt tbk p`rtfgok tìks t tfdk tn rk`gb

(l) @vkr`jk vkongfty

4

4

Mkt hfspo`gkdkmt hfspo`gkdkmt Xfdk Xfdk tìkm tìkm

∜ 15 d 1.0s

4 ∜ 6.2 d/s 6.2 d/s (hnwmw`rhs) 4 6.2 d/s Χ vt vkongfty 4 8. @vkr`jk vkongfty Χt vt + < vt5 + 7vX <.;v <.; v 4 5 t5 + t5 + X ;t5 + <.;X 4 7 t5 + 7 X ∸ nr X 4 8 t5 ;. (`) @vkr`jk `ggkokr`tfnm

4 4

Afmò Afmò vkon vkongf gfty ty ∜ fmftf fmftfò ò vkon vkongf gfty ty (8

+ 2) s

5∜5 0<

4 5 d/s< v (g) ` 4 t`m β 4 d`x 8

v v d`x

jrnumh. +<5 d/s ‗ 15 d

b

β 8s

2s

0

















(l) @s tbk p`rtfgok p`rtfgok hfh mnt rkturm l`gi hfst`mgk tr`vkookh fm 0< s

1.

4 Hfspo`gkdkmt `t 0< s ∸ @vkr`jk spkkh 4 2 d/s. <0 (`) \` \`hfus ( \) na gfrgok 4 d

:. Unsftfnm vkgtnr `t t 4 5 s ↜ ]

r0 4 (0 f

∸ Gfrgudakrkmgk na gfrgok 4 <υ \ 4 < Ú << Ú <0 d

(`) Hfspo`gkdkmt arnd t 4 5 s tn t 4 8 s ↜ ↜ 4 ( r< ∜ r0 )

<<

4 (1 ]f + 8 ]c) ∜ (0 ]f + < ]c)

4 1d Wpkkh (v) na p`rtfgok

∸

Hfst`mgk dnvkh dnvkh ly p`rtfgok fm < s 4 < d

Xbus, `mjok tbrnujb wbfgb tbk p`rtfgok dnvkh 4 < Ú <υ 4 <υ 4 0<5µ 1 7 D`jmftuhk na @vkr`jk vkongfty hfspo`gkdkmt 4 D`jmftuhk na hfspo`gkdkmt Xfdk ( 4 < s)

(t 4
0<5µ N

↜ vL

4 @L 4 <\ sfm 15µ <

4 \ 7 <

<

∸

@vkr`jk vkongfty 4

]

(; f

+ < ]c) d 8s

4 (0.<; ]f + 5.; ]c) d/s (l) @vkr`jk `ggkokr`tfnm Afmò fmò vkon vkongf gfty ty ∜ Fmft Fmftf` fòo vkon vkongf gfty ty

4 4

@( @(t 4 5 s) L

4 (; ]f + < ]c) d

4 0 d/s

↜ v @

+ < ]c) d

Unsftfnm vkgtnr `t t 4 8 s ↜ ] ] r< 4 (1 f + 8 c) d

<<

:

<0

4

8s ]

(< f

+ 05 ]c) ∜ ( 8 ]f + 1 ]c) 8

∜ < ]f + 8 ]c 8

] ] 4 ( ∜5; . f + c) d/s<

(g) Zk g`mmnt afmh tbk `vkr`jk `vkr`jk spkkh `s `s tbk `gtuò p`tb anoonwkh ly tbk p`rtfgok fs mnt imnwm. [mfanrd `ggkokr`tfnm (`) Nmk hfdkmsfnmò dntfnm

















<< | Dkgb`mfgs-0 v5 t ∜

0 < jt < 54

nr

4 v5 ( t ∜ t5 ) ∜ 0 j( t ∜ t5 )<

∜ s 4 ( `t5 ) t + 0 ( ∜ `) t<

<

<

∜ v5 t5 ∜ 0 j( t<5 ∜
< 0 < jt5 t 4 jt5 + v5 t5 < 0 jt5 + v5 < t4 j 4 t5 + v5 j < 0 ; 4 jt< <

nr

nr

05.

⇜

∸

∜ 0 `t<5 4 ( `t5 ) t ∜ 0 `t<

nr

∜ 4
t4

4

@ N B Z

< <

(∜ fvk

L u 4 5 ( ‗0) t skgnmh anr L

0<.

⇜

;< u<

0 5)0; 4 u< + < ( ∜05 4 7<; v 4 4 5

@

L

; d/s B

Anr @ B

4 5. t + 0 jt<

B

4

0;d

< 0 < jt <

…(f)

0

<

< 0

4 u< + <( ∜05) B <5 <5 B 4 4 7<; B 4 . d 4 01<; 5<

∜ <; 4 0 j( t ∜ 0)< (

)<

…(ff)

u

(`) Anr B

Anr L B

lkfmj

Arnd tbk lkjfmfmj na tbk dntfnm tbk pnfmt d`ss wfoo rkturm tn tbk fmftfò pnsftfnm àtkr tfdk 7080 . t5 .

\

nr

sfjm

4 (0 + <) t5 t5 4 <080 .

t skgnmh skgnmh anr @ anr @

K

< <

°
4 t5 + t5

0s <; d

<

`lsurh)

Xfdk fmtkrvò

;d

<

t<5

nr

t 4 0 s X

<

f.k., f.k., nr

(l) Anr t


















∜ <5 u + :; 4 5 (u ∜ 0;)(u ∜ ;) 4 5 u 4 ; d/s

u<

f.k.,

∸

` 4 + x

08. Wt`rts @

` 4

7

t 0

d / s<

4 5 + xt0 v
vd`x `mh

⇜

s 4

4 4

0 < `t < 0 ; Ú < 7 ; < t 1

t(s )

5

v (d (d/s)

5 0<5

5 4 vd`x `mh

Ú t<

⇜ …(fff)

;/1

7

1

:.;

6

75 1:.; 0<5

+ ( ∜ y) t<

4 < ys< v t< 4 d`x

v
∸

vd`x x

nr

0 vd`x   x

0<

+

y vd`x y

65 15 75 0; 1

6

0<

t (s)

Hfaakrkmtf`tfmj Kq. (fff) w.r.t. tfdk t

…(fff) …(fv)

4 t0 + t< 4 8

+ 0  4 8 y

Arnd Kq. (ff) `mh Kq. (fv) v
s (d)

7

…(ff)

Cnurmky Cnurmky U tn L

4 :.; d

0

…(f)

x

7

(h)

s<

Cnurmky Cnurmky @ tn U

4 5< + <`x u< (;)< x 4 4 <` < Ú ;

f.k.,

Wtnps L L

8 dfm

u<

(g)

t <

s0

(l) [sfmj Kq. (ff) ;

` 4 ‗y

8 id

(0; d/s (0; d/s lkfmj mnt pnssflok)

<7

…(v)

…(vf)

Hfvfhfmj Kq. (vf) ly Kq. (v) vd`x 4 < Wulstftutfmj tbk vòuk na vd`x fm Kq. (v) 0 0 (Urnvkh) + 4 < x y

















<8 | Dkgb`mfgs-0 (l) v(`t t 4 05 s)

4 u + ` 05 4 ∜ 1` + 05` 4 8` 4 52. d/s

(g) Xwn nr tbrkk hfdkmsfnmò dntfnm ↜ ↜ A 05 M mnrt mnrtb b `4 01. 4 d < ij

↜ u ↜ `<

↜

4 5 d/s

4 ∜ 8 ]c d/s< ( t 4 < s tn t 4 8 s) t< 4 < s

(`) Qkongfty ↜ v0

↜

4 8 ]f ↜ v<

4 05 f d/s ↜ v

4

↜ v<

nr

]

↜ u

↜

↜

4 (2 ]f ∜ 2 ]c ) d/s

(l) Gn-nrhfm`tk na p`rtfgok ↜ ↜ ↜ 0↜ s 0 4 s 5 + u t0 + ` 0 t0< <

↜ +`t

4 05 ]f + (; ]c Ú <)

4 (< ]f + 8 ]c) + (5)(<) + 0 (< ]f) << <

4 05 ]f + 05 ]c ↜ |v|4 05 < d/s ↜ v 4 05 <, mnrt mnrtbb-k` k`st st ↜ ↜ 0↜ usfmj s 4 u t + ` t< < 4 (05 ]f Ú <) + 0 (; ]c)<< <

4 (<5 ]f + 05 ]c) d ↜ |s| 4 <5<

↜

4 v0 + (` 0 + ` < ) t< 4 8 ]f + (< ]f ∜ 8 ]c ) <

4 05 d/s, k`st

↜ usfmj v

↜

4 5 + (< ]f) <

4 ; d/s< , mnrtb 4 ; c] d/s< ↜ u

↜

4 u + ` 0 t0

+ 05<

4 05 ; d <5 gnt β 4 <

4 < ]f + 8 ]c + 8 ]f 4 1 ]f + 8 ]c ↜ s<

4

↜ s0

↜

+ v0 t< + 0 ( `0 + `< ) t<< <

4 1 ]f + 8 ]c + ( 8 ]f) < + 0 (< ]f ∜ 8 ]c) << <

4 1 ]f + 8 ]c + 2 ]f + 8 ]f ∜ 2 ]c 4 02 ]f ∜ 8 ]c Gn-nrhfm`tk na tbk p`rtfgok Y02 d , ∜ 8 d T




















4 05 ]f ∜ < ]c

06.

4

∸ Gn-nrhfm`tks na p`rtfgok wnuoh lk Y05 d , ∜ < d T ↜ ] u 4 2 c d/s ↜ ] ` 4 (8 f ↜ ↜ s5 4 5

<6 <6 f

;

]

] <

f.k., β 4 t`m0( 8) Mnm umfanrd `ggkokr`tfnm.

4 < + t< +
<0. x

(l)

] <

ct

]

4
<6

⇜

+ <5<

4 <51. d/s <5 t`m β 4

hx ht

4
]

Gndp`rfmj tbk gnkaafgfkmts na f `mh c `mh

55< .

↜ |` `v|4 ;<

+ < ]c) d/s<

+ m c 4 2 ct +
+ 5.8 ]c

4 ; ]f + <5 ]c d/s

↜ ] ] s 4 <6 f + m c ↜ ↜ ↜ 0↜ (`) s 4 s 5 + u t + ` t< < ↜ 0 ] ] ] ] ] <6 <6 f + m c 4 5 + (2 c) t + ( 8 f + < c) t< < ]

]

5.0 f

…(f) (g)

…(ff)

( `) t 4 < s

<

Wulstftutfmj vòuk na t fm Kq. (ff)

h< x ht<

<<.

4 < + 0
4 < + (0< Ú <) 4 <1 d/s< hv `4∜v

<;

















<1 | Dkgb`mfgs-0 (`) Hfspo`gkdkmt `t 4 Hfspo`gkdkmt `t t 4 5 s 54

⇜

v5

<

7.; <

7

t 4 ; s

∜ ; + v5 ; 7

. s 4 ;/1 4 5277

(l) v 4 7 t ∜ t

+ v5 Qkongfty `t t ( 4 ;55s . ) < 4 7.; ∜ ; + 5.277 4 0; ∜ <; + 277 . d/s 4 ∜ 601: <;. v 4 7 t< ∜ 1t ∸ ∩ hs 4 ∩ (7 t< ∜ 1t) ht nr s 4 t 7 ∜ 7 t< + g nr s 4 t 7 ∜ 7 t< 4 t< ( t ∜ 7) (7.;)< (7.; ∜ 7) (`) @vkr`jk vkongfty 4 <

Mnw, `t t 4 < s, v 4 1 d/s 1< ∸ 4 ( 8 Ú <) + G < Xbus, G 4 05. 05.

v<

f.k., f.k.,

∸

<

…(ff)

t 4 7 s,

`t

<

v <

4 ( 8 Ú 7) + 05

4 88 v 4 88 4 <

v<

00 d/s

Wulstftutfmj `lnvk anumh vòuk na v fm Kq. (f), < 00 ` 4 8 ` 4

f.k., f.k.,

7.;

4 7.; Ú 5.; . d/s 4 0:;

4 8 t + 05

< 00

4 5.157 d/s < <:. @ggnrhfmj tn qukstfnm, tbk vkongfty na tbk

p`rtfgok v`rfks `s sbnwm fm afjurk.

(l) Hfst`mgk gnvkrkh

v <5 d/s


















<:

Jr`pby <2.

N@ 3 sonpk fs + fvk `mh fmgrk`sfmj. s

75.

G L

G`sk F v

β

H

@

s

β

d 4 t`m

⇜

t

s5 t

v 5 `

t

N

` 4 t`m β

∸ vkongfty fs + fvk `mh fvk `mh `ggkokr`tfnm fs + fvk. fvk. @L 3 @L 3 sonpk fs + fvk `mh fvk `mh gnmst`mt ∸ vkongfty fs +fvk `mh `ggkokr`tfnm fs zkrn. LG 3 LG 3 snpk fs + fvk `mh fvk `mh hkgrk`sks. ∸ vkongfty fs + fvk `mh fvk `mh fmgrk`sfmj.

⇜

v 4 dt ∜ v5 ( st ( st-ofmk) -ofmk)

∸

GH 3 GH 3 sonpk fs ‗fvk `mh fmgrk`sfmj

∸ vkongfty fs ∜ fvk `mh `ggkokr`tfnm fs ∜ fvk. <6.

v

∩ hs 4 ∩ (dt ∜ v ) ht 5

f.k., f.k., s 4 d ⋋ nr

s 4 s5

<

t <

∜ v5 t +

v D <

β

D 0

Aurtbkr,

∜ v5 t + s5

`

4

hv ht

0 < `t <

4 t`m β

(U`r`lno`) ( d 4 `)

















<2 | Dkgb`mfgs-0 G`sk FFF.

Xfdk-`ggkokr`tfnm Xfdk-`ggkokr`tfnm jr`pb s

v v 5

tfdk

s5

⇜ β t

t

5-
sonpk

`ggkokr`tfnm ; d/s < 5 d/s < ∜ ; d/s < ∜ ; d/s < + ; d/s <

; zkrn ∜ ; ∜ ; +;

` `

⇜

;

t

` 4 t`m β

v 4 v5

∸

+ dt ∩ hs 4 ∩ (v5 + dt) ht s 4 v5 t

+d

8

1

2

05

t

t<

+ s5 < 0 nr s 4 s5 + v5 t ∜ `t< < hv Aurtbkr, ` 4 4 t`m β ht f.k., f.k.,

<

@s anr 65µ ? β ≨ 025µ

( d 4

∜ `)

70. ` 4 sonpk na v- t jr`pb. W 4 `rk` umhkr v ∜ t jr`pb. Gnrrkspnmhfmj jr`pbs `rk hr`wm fm tbk `mswkr sbkkt. Mkt @ @


















0  ∸ Qkongfty na 0 w.r.t. < `t tfdk t   4 s  <  0 4 ( ∜ <;) + (5)    < 4 ∜ <; d/s <; d/s

5 4 ( + 05) ;t<

nr

↜

↜

↜

[sfmj W 4 W 5

↜

0↜ ` rko t< , <

72.

7.; 7. ; d/s <

`s `t tfdk t (4 tfdk na gnoofsfnm na tbk p`rtfgoks) tbk rko`tfvk hfspo`gkdkmt na < ↜

71.

65

Wpkkh na ksgò`tnr

4

O 15

tfdk 4t 4t s

tfdk tfdk 4 5 <.< <. < d/s < X

X

@

↜

4 N) ↜ 0 N 4 ( ∜ <5) + ( ∜ <;) ⋋     < ⇜ t 4 5.2; s 5.2; s Okt okmjtb na ksgò`tnr O ( O (4 0; d) O wòifmj spkkh na d`m 4

w.r.t. 0 wfoo lk zkrn ((f.k. f.k.,, W

∜ 01t ∜ 05 4 5 t 4 7.1; s

4 ; + (< Ú 7.1;) 4 0<.75 d okvko

4 ∜ <5 d

+ u rko t +

0 ( +05) t< <

Unsftfnm na kokv`tnr wbkm ft dkkts lòo

(h) Fmftfò rko`tfvk hfspo`gkdkmt na < w.r.t. 0 W5

+ ( ∜01) t +

<6

@ x 15 d

(`)

0 (<.<) t< <

⇜ 0 (l) (7.;) t< <

4 15

…(f) t 4 :.76 s

4 15 + x

Hfvfhfmj Kq. (ff) ly Kq. (f), 15 + x 7.;

…(ff)



















75 | Dkgb`mfgs-0 @

85.

< sfm ( 8;µ ∜ β)

L N@L 4 N@ 4 <5 d

<5 d N

8<.

NL 4 (N@)

4

(<5)<

4 <5

+ ( @L)

8 sfm 8;µ

0  β 4 sfm ∜0   ∜ 8;µ  < <   

<∓< d/s 8;µ

<

4

Okt pfont bk`hs pnfmt \ pnfmt \ tn tn rk`gb pnfmt P

<

\

Mnrtb P

+ (<5)<

Zfmh

Wpkkh ònmj NL fs NL fs < < d/s

∸ Xfdk tìkm tn rk`gb L 4

<5 < d < < ds ∜0

55 id/b

β U Wnutb

0555 id K`st


















⇜

X <

4

ud

+

( dj

a ) ( dj dj

∜

a )

…(ff)

2.

[sfmj Kq. (f) `mh Kq. (ff)

⇜

X < X 0

+ 4 dj ∜

X<

= X 0

dj

Anr afrst nmk skgnmh 0 <4 ` <

` 4 8 d/s<

a

⇜

a

Qkongfty `t tbk kmh na mkxt skgnmh v 4 ( 8)

@mjuo`r spkkh (φ ) na skgnmhs b`mh <υ 4 r`h s ∜0 15

Nptfnm (l) fs gnrrkgt. 6.

Wpkkh na tbk tfp na skgnmhs b`mh

υ

0

gd/s

Ú (<)

4 2 d/s

Nptfnm (g) fs gnrrkgt. 7.

Ú 0<

Y

`mh

x4

∜ 7 t + t7

∸ hfspo`gkdkmt `t tfdk t (4 0 s) 4 ∜ 7(0) + (0) 7 4 ∜ < d

70

















7< | Dkgb`mfgs-0 00.

0 j (
+ 1) d (jfvkm)

v<

Nptfnm (l) fs gnrrkgt.

7rh hrnp 0 jt <

<

tfdk 4 t

07.

v<5

4 < jb `mh u< 4 < j (7 b) u< ∸ 47 < v5

nr

0 j (
tfdk 4 t

v x 4 2 t ∜ <

∸ 0st hrnp

0

jt<

Jrnumh

;

d

u 4 v5 7

Nptfnm (`) fs gnrrkgt. 08.

∸

4 < ]f + 8 ]c

nr nr

hx ht

4 2t ∜ <

∩ hx 4 ∩ (2 t ∜ <) ht x 4 8 t<

∜

















@t t 4 5

v

<5.

hy ht

4 2 `mh

hx ht

v d`x d`x

41

∸ Qkongfty na prnckgtfnm <

<

hx   hy  4   +   4  ht   ht  4 05 ds ∜0

0:.

Nptfnm (g) fs gnrrkgt.
∸

X 4 7

<  B     6 

1<

+ 2<

t 0

t

t <

Fm tfdk fmtkrvò t0 @ggkokr`tfnm @ggkokr`tfnm Fm tfdk fmtkrvò t< \kt`rh`tfnm 4

4

vd`x t0

vd`x t<

4`

4 <`

77

















78 | Dkgb`mfgs-0 nr nr

∸

0 `t 4 5 <
4 4

` 1u< `

+

05 @0 N

0

@< <

t

‗ 05

`

Nptfnm (`) fs gnrrkgt. <<.

v (d/s) (d/s)

u 4 vkrtfgò spkkh (w.r.t. g`rt) na tbk p`rtfgok.

∸

Hfspo`gkdkmt na p`rtfgok `t tfdk (t 4 <)

4 ( + ; d ) + ( ∜; d ) 4 5 d. f.k., f.k., tbk p`rtfgok grnssks fts fmftfò pnsftfnm `t t <




















Xbus, ∜

0 v

4 ∜ 5.
0 05

Anr vkongfty v `t tfdk t (4 < s) 0 0 ∜ 4 ∜ (5.< Ú <) ∜



Wulstr`gtfmj Wulstr`gtfmj Kq. (ff) arnd Kq. (fff), 0 b ∜ B 4 u ( t ∜ X) + j ( X< ∜ t< ) < 0 4 jX ( t ∜ X) ∜ j ( t ∜ X) ( t + X )

7;

















71 | Dkgb`mfgs-0

4 0<.; ∜ 01.5 + 05 4 1.; d/s

Nptfnm (l) fs gnrrkgt. 78.

Wkk `mswkr tn qukstfnm mn. <. dj Xfdk na `sgkmt

a

Xfdk na hksgkmt

a

4

∜ dj +

Nptfnm (l) fs gnrrkgt. 7:.

<

< jB


















CKK Gnrmkr @sskr @sskrtfnm tfnm `mh \k`snm \k`snm 0.

↜

@s `ggkokr`tfnm `ggkokr`tfnm (` ) fs cust nppnsftk tn

4

Hfspo`gkdkmt

7:

















72 | Dkgb`mfgs-0

1.

\k`snm fs aòsk `s wbkm `ggkokr`tfnm fs pnsftfvk tbk spkkh wfoo fmgrk`sk. h` 4 < (ds ∜7) (jfvkm) ht

Xfdk t \ko`tfvk Hfspo`gkdkmt 5s 0s
5d 8d <8 d


















W0

0 < jt < 0 b 4 j < j b

4

0<.

76

Uok`sk skk `mswkr tn qukstfnm mn. < na nlckgtfvk qukstfnms (okvko 0). ud X < 4 ( dj + a ) ( dj ∜ a )











































































































































Motion in One Dimension

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