W ë s X ¢ V ê s½ Û 8 ý ú n ÞI í Ä Ã U Ø< g; c 6 X ¢ 6 ] K ¡t ô p §8 ý Þ¶ ¥X ì Ä T A 0Ê ÝX N Ë
» ¸# Ò
∗Ø
æ· ¡ ¤ @ / < Æ § õ < Æ §¹ ¢ ¤ < Æ Ò, ' õ AÅ Ò 361-763
ø ¶ B4 w H
†
z @ / < Æ § õ < Æ §¹ ¢ ¤ < Æ Ò, F g Å Ò 500-757 (2008¸ 5 Z 4 8{ 9 ~ à Î6 £ §)
"
f Ð ² ú Ðs H & ³ © s [ O " î 1 p x` ¦ : x ½ + Ë& h Ü ¼ Ð s K H כ É r Ó ü t o < Æ& h ¦ü < ½ Ó_ & h ¦\
"
f × æ כ ¹ô Ç ô Ç 8 £ ¤ s . Õ ª Q ´ ú § É r ½ ¨\ " f < ÆÒ q t_ > h¥ Æ & h s K B Ä º © S ! _ > r& h s H כ s
Ð ¦÷ &# Q M ® o . : r ½ ¨\ " f H > á ¤ ° ú s × æ§ 4 s 6 x t ë ß " f Ð ² ú Ðs H î r1 l x` ¦ H 4 t â Ä º
\
¦ ] jr ¦, \ Vq Ó ü t o § \ ¦ @ / © Ü ¼ Ð Å Ò# Q 4t © S ! \ @ /K " f : x ½ + Ë$ í ` ¦ d H t , : x ½ + Ë$ í ` ¦
d ô Ç â Ä º\ # Qb G> K 9 ¦ H t \ ¦ ¸ % i . ¸ õ : x ½ + Ë$ í _ d ÷ r m : x ½ + Ë$ í
`
¦ K l 0 Aô Ç õ & ñ s B Ä º < Ê` ¦ · ú Ã º e % 3 . s \ : r ½ ¨ [ þ t É r > h¥ Æ ¸_ Ö ¸6 x õ “ë ß { 9 , ... s ”s H ¦> e _ Ö ¸6 x` ¦ ] jî ß % i .
PACS numbers: 01.40.Fk
Keywords: Óüto&h ¦, :x½+Ë$í, ©S!_>r$í, ½Ó_$í, Óüto<Æ_þvt¸, >h¥Æ¸, ¦>e
I. " e  ] Ø
© S ! (context)\ @ /ô Ç ª ô Ç & ñ _ ü < < Êa [1,2], õ < Æ
§¹ ¢ ¤ \ " f © S ! _ × æ כ ¹$ í \ @ / # ´ ú § É r ½ ¨ [ þ t s /
å
L % i [2–4]. © S ! \ @ /ô Ç / å L É r õ < Æ §¹ ¢ ¤3 l q ³ ð [5], { 9
© & h © S ! \ " f_ Ó ü t o < Æ_ þ v t ¸ [6,7], Ó ü t o ë H ] j K õ
& ñ \ " f © S ! _ % i ½ + Éõ % ò ¾ Ó [8], Õ ªo ¦ < ÆÒ q t > h¥ Æ _ © S !
_ > r$ í \ @ /ô Ç כ [4,9–11]1 p x ª ô Ç ì r \ " f e # Q M ® o
. < ÆÒ q t > h¥ Æ _ © S ! _ > r$ í \ @ /ô Ç \ V Ð H, ë H ] j © S !
\
] jr ) a j Ë µ_ 7 á x À Ó\ < ÆÒ q t[ þ t _ > h¥ Æ s Ø Ô H
Ð ¦ [4], 1 l x{ 9 ô Ç ë H ] j © S ! \ @ /ô Ç s K ² D G ç ß s
\
¦ Ð H ½ ¨ õ [9,10], Õ ªo ¦ { 9 © & h © S ! õ õ
< Æ& h © S ! \ 6 £ x ² ú s Ø Ô H ½ ¨ Ð ¦ [11] 1 p x s e
. Õ ª Q Ó ü t o < Æ_ þ v \ " f © S ! \ @ /ô Ç × æ כ ¹$ í \ q K 5
Å x Ö æ É r s ì r \ @ /ô Ç ' a d õ ½ ¨ Ø æì r t · ú §
¦ t & h % i [2].
< ÆÒ q t[ þ t _ õ < Æ> h¥ Æ \ @ /ô Ç t 5 Å q& h ½ ¨ õ \ _
, < ÆÒ q t[ þ t s ` ¦ É r õ < Æ> h¥ Æ ` ¦ S \ 1 p q # ô Ç t õ ]
j\ @ / # ¸ ú [ O " î % i ¦ 8 ¸ Ä » ô Ç õ ] j\
&
h
6 x < Ê\ e # Q # Q 9¹ ¡ §` ¦ t ¦ e H כ s µ 1 ß) & [12–15]. s Qô Ç < ÆÒ q t[ þ t _ 6 £ x ² ú : £ ¤f ç É r õ < Æ < Ê É r õ < Æ
∗E-mail: [email protected]
†E-mail: [email protected]
§ [ þ t s ª ô Ç ë H ] j © S ! \ @ / # { 9 ' a ) a [ O " î ^ >
\
¦ t ¦ [ O " î H כ õ É r : £ ¤f ç s . 1 l x{ 9 ô Ç Ó ü t o > h
¥ Æ
Ü ¼ Ð [ O " î ½ + É Ã º e H Y > > h_ ë H ½ Ó\ @ / # õ < Æ §
<
Ê É r õ < Æ ü < ² ú o , < ÆÒ q t[ þ t É r s [ þ t ë H ] j\ @ / # 1 l x{ 9 ô
Ç ë H ] j Ð d t 3 l w 9 _ > h¥ Æ Ü ¼ Ð : x ½ + Ë ) a [ O
"
î ` ¦ l Ð H y y _ ë H ½ Ós ° ú H : £ ¤f ç 7 £ ¤, ] jr ) a ë
H ] j © S ! \ Ø Ô> 6 £ x ² ú H â ¾ Ós e # Q Ð .
¢
¸ô Ç < ÆÒ q t[ þ t s ] jr H [ O " î \ " f § < Ê É r ½ ¨ [ þ t
É
r 7 H o & h ¸í H` ¦ t t ë ß & ñ < ÆÒ q t[ þ t É r s Qô Ç ¸ í
H` ¦ t t 3 l w # © S ! \ É r y y _ s K 1 l qw n
&
h Ü ¼ Ð { © ¦ Ò q ty # [16–18], _ > h¥ Æ ` ¦ { 9
'
a ÷ &> + þ A$ í t 3 l w H â ¾ Ós µ 1 Ï| ÷ &% 3 [14,19]. ¢ ¸ ô
Ç < ÆÒ q t[ þ t s ë H ] j © S ! [ þ t s _ Ä » & h ` ¦ d H X <
¸ © { © ô Ç # Q 9¹ ¡ §` ¦ t ¦ e 6 £ §` ¦ µ 1 ß+ À I [20]. 0 Aü < ° ú
É
r : £ ¤f ç [ þ t ÐÂ Ò' < ÆÒ q t[ þ t s © S ! \ ² ú o Ðs H s
\ ¦ F G4 ¤ ¦ { 9 ì ø Í& h Z O g Ë :` ¦ s = å J # Q? / H 0 p x§ 4 ` ¦ µ 1 Ï
² ú
r v l 0 Aô Ç | Ä Ìs 9 כ ¹ ¦ ½ + É Ã º e [14].
¢
¸ô Ç ~ à Ì7 á x" é ¶ [5] É r õ < Æ& h ½ Ó_ $ í ` ¦ µ 1 Ïí ß & h ¦, à º§ 4
&
h ¦, Õ ªo ¦ ' a& h ¦ Ð & ñ _ " f, à º§ 4 & h ¦ _
ô Ç כ ¹ è Ð : x ½ + Ë$ í ` ¦ í < Êr e . 7 £ ¤ : x ½ + Ë$ í _
d õ כ ¹½ ¨ H Ó ü t o < Æ_ ½ ¨÷ r m õ < Æ& h ½ Ó_ $ í
\
" f ¸ × æ כ ¹ô Ç ¦ ¦ ½ + É Ã º e . z ´Y V Ð Ã »
É r 16 [ j M : y n C_ 5 Å q ¸ Ð ² ú o " f y n C` ¦ Ð ¤` ¦ M : # Qb G>
-475-
Ð{ 9 t \ ¦ ¦z ´+ « > " f, ° ú w n =Y U ¸_ © @ / : r õ } Û ¼R / ÷ _
1 l x s : r s : x ½ + Ë& h Ü ¼ Ð { 9 ' a ) a [ O " î ` ¦ t 3 l w ô Ç
H כ ` ¦ d ¦, Õ ª ë H ] j_ K ` ¦ : £ ¤ Ã º © @ / : r Ü ¼ Ð µ 1 Ï
r e . " f < ÆÒ q t[ þ t \ > : x ½ + Ë& h ¦\ ¦ [ v
H כ s 9 כ ¹ ¦ x .
×
æ§ 4 É r < Æ § §¹ ¢ ¤ õ & ñ \ " f × æ כ ¹ô Ç > h¥ Æ Ü ¼ Ð 2 [/ å L ÷ &# Q
ª ô Ç © S ! \ " f_ × æ§ 4 _ % ò ¾ Ó` ¦ À Ò ¦ e . < ÆÒ q t_ Ó
ü
t o > h¥ Æ ½ ¨ ì r \ " f ¸ ´ ú § É r ½ ¨ [ þ t s ª ô Ç × æ
§
4 © S ! \ " f < ÆÒ q t_ ¸> h¥ Æ ` ¦ ½ ¨ % i . \ V\ ¦ [ þ t , f
î r1 l x [4,21–26] s " é ¶ ( " é ¶)î r1 l x [27–33]1 p x, < ÆÒ q t[ þ t s
© S ! \ @ / # { 9 ' a ) a 6 £ x ² ú ` ¦ Ðs t 3 l w ô Ç H כ s Ð
¦ ) a e . " f © S ! [ þ t 5 Å q \ " f × æ§ 4 \ ' a ô Ç { 9 ' a
)
a d ` ¦ [ v l 0 AK §Ã º] X © : £ ¤Z > ô Ç ¦ 9 9 כ ¹
x [14,34].
: r ½ ¨\ " f H × æ§ 4 s 6 x t ë ß Ó ü t ^ _ î r1 l x s ² ú o
Ðs H # Q t © S ! \ " f < ÆÒ q t[ þ t s # Qb G> y © S ! [
þ
t ç ß _ q { 9 ' a$ í ` ¦ d ¦ ² ú Ðs H © S ! [ þ t` ¦ { 9
'
a& h Ü ¼ Ð K ¦ H t \ ¦ ¸ ¦ ô Ç . 7 á § 8
½
¨^ & h Ü ¼ Ð ´ ú , × æ§ 4 s 6 x t ë ß " f Ð É r î r1 l x
`
¦ H 4 t © S ! ` ¦ ] jr # y © S ! [ þ t` ¦ Ó ü t o & h Ü ¼ Ð 1
l x{ 9 ô Ç © S ! Ü ¼ Ð d H t , É r © S ! Ü ¼ Ð d H t
, Õ ªo ¦ " f Ð ² ú Ðs H 4 t © S ! ` ¦ # Qb G> : x ½ + Ë
&
h Ü ¼ Ð s K 9 ¦ H t \ ¦ ¸ ¦ ô Ç . s Qô Ç
½ ¨ H < ÆÒ q t[ þ t Ð # F K © S ! _ > r$ í ` ¦ F G4 ¤ # Ó ü t o < Æ
& h > h¥ Æ ` ¦ ° ú ¸2 ¤ Ã º\ O ` ¦ > S \ ¦ t ¸ H X < l í
&
h ½ ¨ « Ñ\ ¦ ] j/ B N ½ + É כ Ü ¼ Ð l @ /ô Ç .
II. ì Å U ê s0 n É
1. כ r Ç Ã U ØU
"
f Ð ² ú Ðs H © S ! [ þ t` ¦ Ó ü t o & h Ü ¼ Ð # Qb G> : x ½ + Ë r
v H · ú Ðl 0 Aô Ç 4> h_ © S ! s < ÆÒ q t[ þ t \ > ] jr
÷
&% 3 : (1) t ³ ð H % \ " f à ºf z H Ó ü t ^ , (2) t
½
¨ Å Ò0 A\ ¦ " é ¶î r1 l x H / B N 0 A$ í , (3) t ³ ð H % \ " f í Ó
ü t î r1 l x H Ó ü t ^ , (4) I ª Å Ò0 A\ ¦ " é ¶ C ¸ î r1 l x
H ' $ í . y y _ â Ä º H ¸¿ º × æ§ 4 ` ¦ ~ à Πî r1 l x H Ó ü t
^
¦ ] jr % i .
0
A 4> h_ © S ! É r ¸¿ º × æ§ 4 ` ¦ ~ à Πî r1 l x H © S ! s
. " f : r| 9 & h Ü ¼ Ð 1 l x{ 9 ô Ç î r1 l xZ O g Ë :s & h 6 x ÷ &# Q ô
Ç . Õ ª Q t ³ ð H % \ " f_ × æ§ 4 É r ë ß Ä » § 4 ` ¦ H
& h Ü ¼ Ð & h 6 x ô Ç â Ä ºs . 7 £ ¤ ë ß Ä » § 4 \ " f o _ s
_ \ O H H \ ¦ & h 6 x # { 9 & ñ ô Ç j Ë µ (× æ§ 4 )s
6 x ô Ç כ Ü ¼ Ð & ñ ô Ç כ s . ¢ ¸ t ³ ð H % \ " f f î
r1 l x õ íÓ ü t î r1 l x _ â Ä º\ H íl ¸| \ _ K î r1 l x s
Ø Ô> è ß כ { 9 ÷ r Ñ ü t “ Ä »z ” î r1 l x Ü ¼ Ð 1 l x { 9
> ^ ¦ Ã º e H î r1 l x s . ¢ ¸ / B N 0 A$ í _ â Ä º ¸ ' $ í _
î r1 l x õ ° ú s o _ ] jY L \ ì ø Íq Y V H ë ß Ä » § 4 s
6 x Ù ¼ Ð " é ¶î r1 l x{ 9 à º e Ü ¼ íl ¸| \ _ K ¢ ¸
H H & h Ü ¼ Ð " é ¶î r1 l x Ü ¼ Ð Ð ¦ e . ² D G 4 > h_ î r1 l x
É
r : r| 9 & h Ü ¼ Ð 1 l x{ 9 > × æ§ 4 s 6 x H î r1 l x s t ë ß ,
H \ ¦ & ñ , íl ¸| \ Ü ¼ Ð Ðl \
É
r î r1 l x s è ß â Ä º ¦ ½ + É Ã º e . " f 4t î r 1
l
x` ¦ : x ½ + Ë& h Ü ¼ Ð s K l 0 AK " f H : r| 9 & h Ü ¼ Ð × æ§ 4 s
6 x ô Ç H d õ < Êa © S ! \ H ü < íl ¸|
`
¦ & h ] X > s 6 x| ¨ c 9 כ ¹ e (Fig. 1).
s
ü < ° ú s y y ² ú Ðs H 4 t © S ! ` ¦ ] jr ¦
< ÆÒ q t[ þ t \ > | 9 ë H ô Ç ? /6 x É r 2 t % i (Table 1). Q1 É r
$ < ÆÒ q t[ þ t s " f Ð ² ú Ðs H 4 t © S ! \ @ /K " f :
x ½ + Ë_ 9 כ ¹$ í ` ¦ d H t · ú Ðl 0 Aô Ç כ s . s M
: < ÆÒ q t s Ø æì r y K ² ú t H © S ! s | 9 ë H
`
¦ ] jr t · ú §` ¦ à º ¸ e Ü ¼Ù ¼ Ð, “ s ¸Ø Ô H | 9 ë H
¸ a % ~ ¦, É r · ú 8 ¸ × æ1 p x < ÆÒ q t[ þ t s ¸Ø Ô ¦ ½ + É Ã º e
H | 9 ë H ¸ a % ~_ þ v m .” ¦ î ß ? /\ ¦ % i .
Q2 H s ] jî ß ô Ç | 9 ë H[ þ t \ @ /K " f Û ¼Û ¼ Ð # Qb G>
² ú
` ¦ ? /o H t · ú Ðl 0 Aô Ç כ s . : £ ¤ y : r ½ ¨ü <
'
aº K " f H Q1 \ " f : x ½ + Ë$ í õ ' aº ) a | 9 ë H` ¦ ô Ç â Ä º\ Q2 \ @ /K " f # Qb G> : x ½ + Ë& h [ O " î ` ¦ ] jr H t \ ' a d
` ¦ ° ú ¦ ì r$ 3 % i . # l \ " f < ÆÒ q t[ þ t s ] jr ô Ç [ O " î
\
H : x ½ + Ë$ í ` ¦ # Qb G> K ¦ H t ü < < ÆÒ q t[ þ t \
>
: x ½ + Ë$ í ` ¦ # Qb G> t ¸½ + É כ t \ @ /ô Ç ? /6 x s < Êa
í < Ê ) a .
2. ì Å 6 V ê s
: r [ O ë H t \ à Ð# ô Ç < ÆÒ q t É r # 3 @ / õ < Æ §¹ ¢ ¤ < ÆÂ Ò Ó ü t o
/ B N 4 < Ƹ < ÆÒ q t 19" î s % 3 . < ÆÒ q t[ þ t s 6 £ x ² ú H X < è כ
¹ ) a r ç ß É r 30ì r s % 3 .
3. Ä Z ØV Ä
Q1 \ @ /K " f H $ : x ½ + Ë$ í ` ¦ d ô Ç | 9 ë H Ä »+ þ Aõ Õ ª X O
t · ú § É r Ä »+ þ AÜ ¼ Ð ¾ º ¦, r [ j ÒÄ »+ þ AÜ ¼ Ð ì r À Ó
%
i . Q2\ " f H Q1 \ " f : x ½ + Ë$ í ` ¦ d ô Ç | 9 ë H \ @ /ô Ç 6
£
x ² ú ë ß ` ¦ ì r$ 3 % i . 6 £ x ² ú ` ¦ : x K : x ½ + Ë o Ä »+ þ A` ¦ µ 1 Ï|
H X < í& h ` ¦ ´ ú Æ Ò% 3 .
Õ
ªo ¦ Q1_ Ä »+ þ A\ É r Q2 _ Ä »+ þ A` ¦ ì r$ 3 ¦, Q2 \ É r Q1 _ Ä »+ þ A ¸ ì r$ 3 % i ¦, Q1õ Q2\ " f 6 £ x
² ú
s ¸ É r ¦ ô Ç â Ä ºü < É r · ú t ë ß × æ1 p x < Æ Ò q
t[ þ t s ¸\ ¦ כ s ¦ ô Ç â Ä º ¸ q § % i .
Fig. 1. Unification of four motions under the gravitation.
Table 1. Questionnaire.
Type Question
Following four cases are motions under the gravitation.
Q1 Ask any questions about these motions. You can ask several questions.
And you can also give questions which can be asked by secondary school students.
Q2 Suggest any explanations or answers about the questions in Q1.
III. + s Ç Ê Ý
1. Q1 ; c 6 X ¢ ù m Ç Ã U Ø ¥] k ù
19" î _ < ÆÒ q t[ þ t s Q1\ " f ] jr ô Ç | 9 ë H É r 8 ú x 38 > h Ð 1 {
© 2> h% i . Q1\ @ /K " f < ÆÒ q t[ þ t s ] jî ß ô Ç | 9 ë H Ä »+ þ A
É
r ß ¼> 3t Ð ì r À Ó÷ &% 3 : (1) : x ½ + Ë$ í ` ¦ d ô Ç â Ä º (22 > h, 56 %), (2) : x ½ + Ë$ í õ ' aº \ O H â Ä º (14> h, 37 %), (3) ¸> h¥ Æ s í < Ê ) a â Ä º (2> h, 5 %) (Table 2). Õ ªo ¦
r : x ½ + Ë$ í ` ¦ d ô Ç â Ä º H ^ & h Ü ¼ Ð : x ½ + Ë$ í ` ¦ d
ô Ç â Ä º (U1-5> h, 13 %)ü < Â Òì r& h Ü ¼ Ðë ß : x ½ + Ë$ í ` ¦ d
ô Ç â Ä º (U2-17> h, 43 %) Ð Ð ü t à º e % 3 (Table 2).
4 > h © S ! ^ \ @ /ô Ç : x ½ + Ë$ í ` ¦ d ô Ç â Ä º (U1) H 38 > h × æ \ " f 5> h (13 %)% i . U11\ @ /K " f : r s ¸ É r
¦ " f | 9 ë H ô Ç â Ä º 3> h (8 %)% i ¦, Qt 2> h (5 %) H É r · ú t ë ß × æ1 p x < ÆÒ q t[ þ t s ¸\ ¦ Ã º e ¦ ô Ç
| 9
ë H s % 3 .  Òì r& h : x ½ + Ë` ¦ d ô Ç â Ä º (U2)\ H, " é ¶î r 1
l
x õ " é ¶î r1 l x` ¦ q § (U21), z H î r1 l x õ C
¸î r1 l x` ¦ q §ô Ç â Ä º (U22, U22’), Õ ªo ¦ × æ§ 4 õ ë ß Ä »
§ 4 ` ¦ q §ô Ç â Ä º (U23, U24)% i . Â Òì r& h : x ½ + Ë d ë
H ] j\ @ /K " f : r ¸ ¸ É r ¦ ô Ç | 9 ë H É r 8 > h (21 %)% i
¦, 9> h (22 %) H É r · ú t ë ß × æ1 p x < ÆÒ q t[ þ t s ¸\ ¦ כ s
¦ ô Ç | 9 ë H s % 3 .
U1 ¢ ¸ H U2 \ K { © H | 9 ë H Ã º H 8 ú x 22 > h Ð 56 %% i t
ë ß , ^ & h : x ½ + Ë$ í ` ¦ d ô Ç ë H ] j (U1)\ ¦ ] jî ß ô Ç < ÆÒ q t
É r 5" î (13 %)Ü ¼ Ð B Ä º & h É r כ Ü ¼ Ð z ¤ .
l
_ â Ä º\ H = Õ ª î r1 l x` ¦ H t \ @ /ô Ç | 9 ë H (O4, O5, O6, O7), j Ë µ_ 6 x \ @ /ô Ç | 9 ë H (O1, O2, O3, O8, O9, O10), D h Ðî r © S ! \ " f # Q " õ \ V8 £ ¤ ÷ & H t
\
@ /ô Ç | 9 ë H (O11)[ þ t Ð" f, : x ½ + Ë$ í õ H © ' a s \ O H | 9 ë
H[ þ t s % 3 .
Q1 \ @ /K " f ¸> h¥ Æ s í < Ê ) a | 9 ë H ¸ e % 3 H X <, M1_
â
Ä º\ H × æ§ 4 mg 6 x # íÓ ü t î r1 l x` ¦ H â Ä º
\
× æ§ 4 mg à ºf A Ð { 9 & ñ > 6 x H × æ§ 4 s m
t ½ ¨ × æd ` ¦ ¾ Ó H × æd § 4 © Ü ¼ Ð ¸ ú 3 l w s K ô Ç â Ä
º% i . z ´ t ½ ¨ ^ \ ¦ Ò q ty t ½ ¨ × æd Ü ¼ Ð ¾ Ó
H כ s ` t ë ß , ² D G è& h Ü ¼ Ð H mg ° ú É r ß ¼l ü < ° ú É r
~
½ Ó ¾ ÓÜ ¼ Ð 6 x H כ Ü ¼ Ð & ñ ¦ e . 7 £ ¤ ² D G è& h Ü ¼
Ð x − y ý a³ ð> \ ¦ [ O & ñ −y ~ ½ Ó ¾ ÓÜ ¼ Ðë ß 6 x H כ Ü
¼ Ð 9, r − θý a³ ð> \ ¦ [ O & ñ # −r ~ ½ Ó ¾ ÓÜ ¼ Ð 6 x
H כ Ü ¼ Ð [ O " î t · ú § H . M2_ â Ä º\ H 8 ú x · ú _ î r 1
l
x ¸ ² D G t ½ ¨ × æ§ 4 _ % ò ¾ ÓÜ ¼ Ð íÓ ü t î r1 l x` ¦ l º
X <, 8 ú x · ú _ î r1 l x s f î r1 l x s ¦ ¸ ú 3 l w Ò q ty ô Ç â Ä
º% i .
2. Þ¶ ¥× D ¥] k ù
:
x ½ + Ë$ í ` ¦ d ô Ç | 9 ë H \ @ /K # Qb G> : x ½ + Ë o\ ¦ r ¸
H t \ ¦ · ú Ðl 0 AK Q2\ @ /ô Ç 6 £ x ² ú ` ¦ ì r$ 3 # & ñ o
% i . s M : Q1 × æ \ " f U1õ U2\ K { © ÷ & H | 9 ë H` ¦ ô Ç
Table 2. Types of questions in Q1 (N=38*).
Type Questions Number
U1: Questions recognizing overall unification 5 (13 %)
U11 Why are all motions different even though all move under the gravitation? 5 (13 %)
U2: Questions recognizing unification partly 17 (43 %)
U21 Satellite moves circularly but planets move elliptically. Why? 8 (21 %) U22 Some objects fall eventually, but satellite and planets do not, why? 3 (8 %)
All objects in the Earth will fall because of gravitational force, U22
0but satellite does not fall. Why? 2 (5 %)
Gravity acts near the Earth, but universal gravitation acts on the sun and planets.
U23 Then, are gravity and universal gravitation different with each other? 3 (8 %) In two cases, gravity varies according to the height, but remains are inversely
U24 proportional to the square of the distance. What make these different? 1 (3 %)
O: Other questions 14 (37 %)
It looks like that there is no direct relationship between the direction of force
O1 and the direction of motion. Why? 1 (3 %)
O2 Can we think other forces besides a gravitation? 1 (3 %)
Speed of falling object is increasing due to gravity, but after some time,
O3 speed will be constant due to air friction. Then, students can ask whether 1 (3 %) gravity acts only before attaining a constant speed.
O4 Why does speed of planet vary according to the area in Kepler’s law? 2 (5 %) Why does satellite move in orbital? Is it possible to be deviated from the
O5 orbit in the place where gravity is weak? 2 (5 %)
O6 Why is the orbit of satellite is always circle? Is non-circular orbit possible? 1 (3 %)
O7 Why do the planets always move elliptical orbit? 2 (5 %)
O8 When object falls due to gravity, then can the Earth move also? 1 (3 %)
O9 Why does gravity act to only y-direction? 1 (3 %)
O10 Is it possible that Sun collides with the planets? 1 (3 %)
O11 If a satellite is separated into two parts, then will the orbits of them be unchanged? 1 (3 %)
M: Questions including misconceptions 2 (5 %)
The direction of gravity should be centripetal.
M1 But why is the direction of gravity downward direction? 1 (3 %) M2 Bullet of gun flies in straight line, then does not the gravity act on the bullet? 1 (3 %)
* The number of students was 19, but the number of responses was 38 because some students asked several questions.
â
Ä º\ ë ß 6 £ x ² ú ` ¦ ì r$ 3 % i . : x ½ + Ë o Ä »+ þ A É r ß ¼> 2t
Ð ì r À Ó½ + É Ã º e % 3 : (1) " f Ð s H כ Ü ¼ Ð Ðs t
ë ß : r| 9 & h Ü ¼ Ð H ° ú É r כ s , (2) " f Ð s H
כ
É r " f Ð É r â Ä ºs l M :ë H s (Table 3).
· ú
¡" f / å L ô Ç ü < ° ú s , : r ½ ¨\ " f ] jr ô Ç 4> h_ © S !
É r ¸¿ º × æ§ 4 s 6 x H © S ! \ " f_ Ó ü t ^ _ î r1 l x s
H & h \ " f : r| 9 & h Ü ¼ Ð 1 l x{ 9 ô Ç â Ä ºs . é ß t " f Ð î r 1
l
x s ² ú Ðs H s Ä » H H \ ¦ 6 x íl ¸| s
Ø Ôl M :ë H s .
: r ½ ¨ õ , 22> h_ 6 £ x ² ú × æ \ " f 14 (64 %)> h_ 6 £ x ² ú s
: r| 9 & h Ü ¼ Ð 1 l x{ 9 ô Ç כ s ¦ ` > d ô Ç כ Ü ¼ Ð
z ¤ . Õ ª Q íl ¸| ë ß Ü ¼ Ð [ O " î ô Ç â Ä º 8" î (36
%) s ¦ H ë ß ` ¦ 6 x # [ O " î ô Ç â Ä º 2" î (9 %)s
%
3 ¦, íl ¸| õ H \ ¦ © S ! \ & h ] X > 6 x
#
4t â Ä º\ ¦ ¸¿ º : x ½ + Ë& h Ü ¼ Ð [ O " î ô Ç â Ä º H \ O % 3
. " f " f Ð ² ú Ðs H © S ! ` ¦ : x ½ + Ë& h Ü ¼ Ð s K
H X < íl ¸| õ H \ ¦ ¸¿ º & h ] X > 6 x H 0 p x§ 4 s
 Ò7 á ¤ < Ê` ¦ · ú à º e % 3 .
8½ ¨ 4" î (18 %)_ 6 £ x ² ú É r " f Ð ² ú Ðs H î r1 l x
É r " é ¶ A : r| 9 & h Ü ¼ Ð É r î r1 l x s ¦ [ O " î < ÊÜ ¼ Ð" f : x
½
+ Ë& h s K _ 9 כ ¹$ í ` ¦ ) d t 3 l w ¦ e % 3 .
Q1 \ " f < ÆÒ q t[ þ t É r s ¸Ø Ô H â Ä º (Table 3\ " f T Ð
³
ðr )ü < É r · ú t ë ß × æ1 p x < ÆÒ q ts ¸\ ¦ Ã º e H â Ä º (Table 3 \ " f S Ð ³ ðr ) Ð ¾ º# Q ] jr % i . < É ª p Ðî r
õ H, É r · ú t ë ß × æ1 p x < ÆÒ q ts ¸\ ¦ כ s ¦ ô Ç â Ä
º\ H 13" î ¸¿ º “ : r| 9 & h Ü ¼ Ð ° ú É r כ s ” H : x ½ + Ë
Table 3. Types of Unification in Q2 (N=22).
Type Unification Number ( %)
S: All motions are the same fundamentally 14 (64 %)
Students suggest explanation that all motions are basically the same.
(e.g.) Circular motion and orbital motion are basically constant falling 4 (18 %) S1 motion like vertically falling motion and projectile motion. S=4
(e.g.) Gravity and universal gravitation are the same with each other. T=0 8 (36 %) S2 (They have same principle, but) they looks (or their motions look)
differently because of different intial conditions or/and intial energy. S=7
T=0 (They are the same, but) they just look differently approximately. 2 (9 %) S3 (e.g.) In fact, gravity is changed according to the height, but it can be seen as S=1
constant because the height is very small compared to the size of the Earth. T=1
D: Each motion is different due to different reasons 4 (18 %)
There are different causes which make four motions different.
(e.g.) Center of mass is located inside the Earth in case of stellite,
but it is located out of the Sun in case of planets. 3 (13 %) D1 (e.g.) In the case of vertically falling object, the direction of motion is S=0
the same with the direction of force, but, in the case of projectile motion, T=3 x-directional motion is constant and y-directional motion is accelerated.
Students do not feel any conflicts between four seemingly
different motions, but regard four cases as different phenomena. 1 (5 %) D2 (e.g.) We can know the reasons why all motions are different S=0
if we use simulation program. T=1
No response 4 (18 %)
o Ä »+ þ A` ¦ Ð% i ¦, s ¸ É r ¦ ô Ç â Ä º\ H @ /Â Òì r (5" î × æ 4" î ) “" f Ð s H כ É r " f Ð É r â Ä ºs l
M :ë H s ” H : x ½ + Ë o Ä »+ þ A` ¦ Ð% i H כ s .
7
£ ¤, < ÆÒ q t s Å Ò# Q 4> h_ © S ! \ @ /K " f s K
Â
Ò7 á ¤ ô Ç â Ä º\ H 4 > h_ © S ! ` ¦ " é ¶ A É r © S ! s ¦ Ð
9 ¦ % i 6 £ §` ¦ · ú Ã º e % 3 . Õ ª\ ì ø ÍK É r · ú ¦ e
¦ ô Ç â Ä º\ H × æ ¦1 p x < ÆÒ q t[ þ t \ > : r| 9 & h Ü ¼ Ð ° ú É r
כ
s ¦ t ¸½ + É כ s ¦ \ V © ½ + É Ã º e . Õ ª Q s â Ä
º\ ¸ H ü < íl ¸| ` ¦ ¸¿ º & h ] X > 6 x ½ + É Ã º e
H 0 p x§ 4 s e ¦ Ðl H j Ë µ[ þ t ¦ x .
s
ü @\ ¸ : x ½ + Ë o õ & ñ \ " f ¹ 1 Ô : £ ¤f ç É r 6 £ § õ ° ú
.
(1) Â Òì r& h Ü ¼ Ð : x ½ + Ë$ í ` ¦ K 9 ¦ Ù þ ¡t ë ß , É r â Ä
º t S X @ / # Ò q ty t 3 l w H â Ä º e % 3 .
\
V\ ¦ [ þ t , < ÆÒ q t A H “' $ í _ î r1 l x \ H ... × æ§ 4 s 6
x K " f " é ¶ C ¸ Ð [ t ¦, ... / B N 0 A$ í É r t ½ ¨\ q K | 9
|
¾ Ós B Ä º ¦, t ½ ¨ s \ e l M :ë H \ " é ¶î r1 l x ô Ç
¦ ^ ¦ Ã º e .” ¦ " f ¸ t ³ ð H % \ " f É r Ó ü t ^
íÓ ü t î r1 l x` ¦ H â Ä º\ H = Ó ü t ^ t ½ ¨\ q K
| 9
| ¾ Ós B Ä º É r X < íÓ ü t î r1 l x` ¦ H t \ @ /ô Ç ¸í H
`
¦ d t 3 l wÙ þ ¡ .
(2) ' aº ) a ? /6 x \ @ /K C î r ? /6 x` ¦ l % 3 # é ß í H
>
& h 6 x % i ` ¦ ÷ r, 7 H o & h [ O " î ` ¦ ] jr t H 3 l wÙ þ ¡ .
\
V\ ¦ [ þ t , 4t © S ! s íl 5 Å q ¸\ É r כ s
H , íl \ -t (° ú כ)\ É r כ s ¦ë ß ½ + É ÷ r,
íl 5 Å q ¸ íl \ -t \ # Qb G> = É r î r1 l x` ¦
H t \ @ /ô Ç ½ ¨^ & h [ O " î ` ¦ ] jr t 3 l wÙ þ ¡ .
(3) : x ½ + Ë o Ä »+ þ A\ " f ¸ ¸> h¥ Æ s í < Ê ) a â Ä º e % 3
.
' Í
P : \ V H, “ / B N 0 A$ í _ â Ä º\ " é ¶ d § 4 õ × æ§ 4 s ° ú
"
f 1 p x5 Å q " é ¶î r1 l x` ¦ ô Ç ”s ¦, ¿ º P : \ V H, “( / B N 0 A$ í _
â Ä º\ ) x ~ ½ Ó ¾ ÓÜ ¼ Ð H 5 Å q§ 4 s { 9 & ñ ¦ y ~ ½ Ó ¾ ÓÜ ¼ Ð
H × æ§ 4 5 Å q ¸\ _ K 5 Å q§ 4 s 7 £ x t ë ß , t ½ ¨ Ñ ü æ/ å J l
M :ë H \ ... " é ¶ C ¸\ ¦ [ t > ) a .” s .
3. Þ¶ ¥V R Ë ß Ã ÅÅ k Ä Ã U Ø< g ¥] k ùÊ Ý Þ¶ ¥× D ¥] k ùÊ Ý8 ý å
¾ Ë4
:
x ½ + Ë$ í ë H ] j Ä »+ þ A\ : x ½ + Ë o Ä »+ þ As ² ú t H t ,
¢
¸ H : £ ¤& ñ ô Ç : x ½ + Ë o Ä »+ þ A\ : £ ¤Z > y ÷ & H : x ½ + Ë$ í ë H
Table 4. Types of unification according to types of ques- tions (N=22).
Question Type Unification Type Number U1 U11 S2 (3), D1 (1), N (1) 5
U21 S2 (3), S3 (1), D1 (2), N (2) 8 U22 S1 (1), D2 (1), N (1) 3
U2 U22
0S2 2
U23 S1 3
U24 S3 1
Table 5. Types of questions according to types of unifi- cation (N=22).
Unification Type Question Type Number
S1 U22 (1), U23 (3) 4
S S2 U11 (3), U21 (3), U22
0(2) 8
S3 U21 (1), U24 (1) 2
D1 U11, U21 3
D
D2 U22 1
N N U11 (1), U21 (2), U22 (1) 4
]
j Ä »+ þ As e H t \ ¦ ¸ ô Ç õ H Table 4 x 9 Table 5ü <
° ú .
Table 4 ü < Table 5\ _ : £ ¤Z > > : £ ¤& ñ ë H ] j Ä »+ þ A
\
: £ ¤& ñ : x ½ + Ë o Ä »+ þ As ) a â Ä º\ ¦ ^ ¦ Ã º H \ O % 3 .
" f y ë H ] j Ä »+ þ A ª ô Ç : x ½ + Ë o Ä »+ þ As 1 l x r \
6 x ÷ &% 3 ¦, y : x ½ + Ë o Ä »+ þ As & h 6 x ) a ë H ] jÄ »+ þ A[ þ t ¸
ª ô Ç כ Ü ¼ Ð z ¤ . 7 £ ¤, ë H ] jÄ »+ þ A\ É r : £ ¤Z > ô Ç : x
½
+ Ë o Ä »+ þ As e , : x ½ + Ë o Ä »+ þ As & h 6 x ÷ & H : £ ¤Z > ô Ç ë H ]
jÄ »+ þ A[ þ t s e H כ É r m % 3 .
IV. + s Ç Â ] Ø õ m Í À X Ø8 ý
: r ½ ¨ õ , < ÆÒ q t[ þ t s " f Ð ² ú Ðs H & ³ © \ @ / K
" f : x ½ + Ë$ í ` ¦ d H â Ä º H 46 % Ð Z } t · ú § ¤Ü ¼ 9, 4 t © S ! ^ \ @ /ô Ç : x ½ + Ë$ í ` ¦ d ô Ç â Ä º H 13 % \ Ô
¦ õ ô Ç כ Ü ¼ Ð , < ÆÒ q t[ þ t s Ü ¼ Ð Ðl \ ² ú Ð s
H & ³ © ` ¦ _ " é ¶ o Ð [ O " î ¦ s K 9 H d s
± ú É r כ ` ¦ ^ ¦ Ã º e % 3 .
#
l \ " f H s Qô Ç : x ½ + Ë$ í _ d ` ¦ [ v l 0 Aô Ç | Ä Ì Ü
¼ Ð > h¥ Æ ¸ (concept map)` ¦ ] jî ß ¦ ô Ç (Fig. 2).
>
h¥ Æ ¸ H ¸~ Ã Ì[6]s 1980¸ @ / ] jî ß ô Ç s Ê ê & ³F \ H > h
¥ Æ
¸ $ í èá Ôà ÔJ ?# Q\ ¦ > hµ 1 Ï # ] j/ B N K Å Ò ¦ e H X <,
#
Q ½ ¨[ þ t \ _ > h¥ Æ [ þ t ç ß _ : x ½ + Ë& h ¦\ ¸
¹
¡
§` ¦ Å Ò H כ Ü ¼ Ð · ú 94 R M ® o .
Fig. 2. Concept Map.
: r ½ ¨\ " f H : x ½ + Ë$ í _ 9 כ ¹$ í ` ¦ d ô Ç Ê ê\ : x ½ + Ë
o\ ¦ 0 Aô Ç | Ä Ì` ¦ ¶ ú ( R Ð ¤` ¦ M :, 64 %_ < ÆÒ q t[ þ t s íl
¸| s H _ 6 x` ¦ : x K : x ½ + Ë& h s K \ ¦ 9 ¦
% i Ü ¼ , Ñ ü t` ¦ < Êa 6 x ô Ç â Ä º H ) \ O H כ Ü ¼ Ð
, : x ½ + Ë& h s K \ ¦ 0 Aô Ç ¦ ^ ¸ B Ä º < Ê` ¦
· ú
à º e % 3 .
Ó ü
t o < Æ H 4 ¤ ¸ ú K Ðs H 5 Å q \ " f { 9 & ñ ô Ç ½ ©g Ë :
$ í
` ¦ ¹ 1 Ô ¦ 4 ¤ ¸ ú ô Ç & ³ © ` ¦ [ O " î H l $ \ ¾ ú 2 ; Z O g Ë
:õ " é ¶ o \ ¦ ¹ 1 Ô ¦ ô Ç . s Qô Ç õ & ñ \ " f Ù þ d & h Z O g Ë
:õ s : r÷ r m , # Q t s © ¸| s 6 x ÷ & [37], Ð ¸[ O s ¸| 1 p x s í < Ê÷ & ¦, íl ¸| s
H 1 p x s 6 x ÷ &l º s . s \ ¦ o & ÐÛ ¼ H õ
< Æs : r s ½ ¨ ¸ o÷ &# Q e ¦ Ð ¦, Ù þ õ Ð ñ@ / Ð s À
Ò# Q ½ ¨á Ô ÐÕ ªÏ þ s H > h¥ Æ ` ¦ ¸{ 9 ô Ç e [38].
" f 1 l x{ 9 ô Ç Ù þ d s : r õ Z O g Ë :s & h 6 xH d \ ¸ Ô ¦ ½ ¨ ¦ Å
Ò# Q Ð ¸s : r õ & ñ , íl ¸| õ H _ 6 x 1 p x \
É r [ O " î õ Ä »+ þ A, & ³ © Ü ¼ Ð l º s . s
Qô Ç & h \ " f õ < ÆZ O g Ë :õ s : r, ¢ ¸ H õ < Æ& h & ³ © ` ¦ s K
H X <\ H Ù þ d s : r õ Z O g Ë :÷ r m , Ð ñ@ /\ @ / ô
Ç s K \ ¦ < Êa 9 כ ¹ Ð ô Ç [39].
Õ
ª Q : r ½ ¨ õ \ " fü < ° ú s \ Vq Ó ü t o § : x
½
+ Ë& h s K \ ¦ 0 AK íl ¸| s H 1 p x` ¦ s K H 0
p x§ 4 s B Ä º ± ú 6 £ §` ¦ · ú Ã º e % 3 . " f s Qô Ç Â Òì r \
"
f_ t ¸\ ¦ 0 AK “ë ß { 9 ... s ”s H ¦ ô ¥º ` ¦ ]
jî ß ¦ ô Ç . s Qô Ç ¦ ô ¥º É r z ´ ´ ú § É r Ó ü t o < Æ
a ¦ H ¦> e Ü ¼ Ð · ú 94 R e # Q [40], z ´] j < Æ Ò q
t[ þ t` ¦ 0 Aô Ç Ó ü t o à º\ O \ " f Ö ¸6 x| ¨ c 0 p x$ í s e ¦ Ò q t y
ô Ç . 7 £ ¤ # Q " [ O " î ^ > s : r, Z O g Ë : 1 p x \ < Êa [ þ t # Q e
H Ð ¸& ñ õ íl ¸| , 6 x ô Ç s © ¸| s H 1
p
x` ¦ or & Ð " f [ O " î ^ > s : r, Z O g Ë :s # Qb G>
² ú
t H t \ ¦ s K H Ö ¸1 l x s . \ V\ ¦ [ þ t , Ä » z
H Ó ü t ^ _ î r1 l x` ¦ s K ô Ç Ê ê\ , “ë ß { 9 z H Ó ü t
^
\ / B N l $ ½ Ós 6 x ô Ç ?” s ¦ ¦\ ¦ S X ©
, “ë ß { 9 Ê êß ¼_ Z O g Ë :\ " f ò ø Í$ í § 4 s Z þ t # Qè ß U ´s _ ] j Y
L \ q Y Vô Ç ?”õ ° ú s { 9 ì ø Í o ) a ¦\ ¦ or & Ð
H ¦ Ö ¸1 l x` ¦ ´ ú ô Ç .
: r ½ ¨\ " f H " f Ð ² ú Ðs H & ³ © \ @ /K < ÆÒ q t[ þ t s
: x ½ + Ë$ í ` ¦ d H t , : x ½ + Ë$ í ` ¦ d H â Ä º\ Õ ª
כ
` ¦ # Qb G> K ½ + É Ã º e H t \ ¦ ¸ % i . : r ¸
H ô Ç t Å Ò] j (× æ§ 4 )\ ô Ç& ñ ÷ &# Q e Ü ¼Ù ¼ Ð : r ½ ¨ õ
\ ¦ { 9 ì ø Í o½ + É Ã º H \ O t ë ß , s ì r _ ½ ¨ B Ä º & h É r
כ
` ¦ Ò q ty ô Ç s ì r _ ½ ¨ü < t ¸\ a % ~ É r r & h
`
¦ × ¦ Ã º e ¦ ó ø Íé ß ) a . É r Å Ò] j\ @ /ô Ç
½
¨ Ð S X © ¦, s Z > Ð ì ø Í6 £ x _ s \ ¦ Ð , { 9 ì ø Í < Æ Ò q
tõ % ò F < ÆÒ q tç ß q § ½ ¨ 1 p x Ü ¼ Ð µ 1 Ï | ¨ c à º e l \ ¦ l
@
/K : r . Õ ªo ¦ : x ½ + Ë$ í _ d õ K ` ¦ 0 Aô Ç ½ ¨^ & h
t ¸~ ½ Óî ß \ @ /ô Ç ½ ¨\ ¦ l @ /K : r .
P
c p 8 ý ò k >
s
7 Hë H É r 2006¸ ¸ Ø æ· ¡ ¤ @ / < Æ § < ÆÕ ü t ½ ¨t " é ¶ \ O _
½ ¨q t " é ¶ \ _ # ½ ¨÷ &% 3 6 £ §. (This work was sup- ported by Chungbuk National University Grant in 2006).
Y
c p w à U Ø ô
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University Students’ Understanding of Various Contexts of Motions with Gravitation as a Unification Process
Ikgyun Kim
∗Department of Science Education, Chungbuk National University, Cheongju 361-763
Jongwon Park
†Department of Science Education, Chonnam National University, Gwangju 500-757 (Received 8 May 2008)
Unifying seemingly different physics phenomena is an important part in thinking and creativ- ity in science. However, literature, has reported that students’ conceptual understanding is very context-dependent. In this study, using four seemingly different motions of objects moving under a gravitational force, we investigated students’ recognition for unification of the four seemingly different cases, and we analyzed how to try to unify them. The students’ level of recognition for unification was found to be low, and their use of solving strategies was not complete. Finally, to help students’ thinking for unification, we propose the utilization of concept mapping and a mental game of ‘what if ...’.
PACS numbers: 01.40.Fk
Keywords: Physics Thinking, Unification, Context-dependent, Creativity, Teaching Physics, Concept map, Mental Games
∗E-mail: [email protected] †E-mail: [email protected]
