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放物運動における最小の到達点速度と 最大の水平到達距離に関する考察

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ௐࠪใࠂ

์෺ӡಈʹ͓͚Δ࠷খͷ౸ୡ఺଎౓ͱ

࠷େͷਫฏ౸ୡڑ཭ʹؔ͢Δߟ࡯

޻౻

༑༟

౦ా

༸࣍

∗∗

The consideration about the initial state with relationship of the minimum speed at a

given point and the furthest hitting point of free falls

Tomohiro KUDO, Yoji HIGASHIDA∗∗

In this paper, leading an angle of projection and initial speed, with holds the minimum speed at a given point of orbit in free falls, is shown. It would be usually shown in algebraically by explaining a horizontal displacement. We show that, it can be calculated by the relation of an angle of projection with the furthest hitting point. It may be useful for understanding of free falls in elementary physics.

Ωʔϫʔυɿࣗ༝མԼ,࠷খ౸ୡ଎౓,࠷େ౸ୡڑ཭Λ༩͑Δڼ֯,ॳ౳෺ཧ

Keywords: free fall, minimum speed, angle of projection with furthest hitting point, elementary physics

1. ·͕͖͑ 2018೥౓ͷߴઐϩϘίϯͰ͸ɺʮϘτϧϑϦοϓɾΧϑΣʯ ͱ͍͏ςʔϚͰɺϖοτϘτϧΛ౤্͛͛ͯߴ͍Ґஔͷςʔ ϒϧͷ্ʹཱͨͤΔڝٕ͕ߦΘΕͨɻϖοτϘτϧΛ౤͛ ͯςʔϒϧͷ্ʹ౸ୡͤ͞Δ·Ͱͷӡಈ͸ɺॳ౳෺ཧͷ์ ෺ӡಈͷԠ༻Ͱ΋͋ΔɻϩϘοτઃܭͰ͸༷ʑͳ࠷దԽͷ ํ๏͕ߟ͑ΒΕΔ͕ɺͦͷதͷҰͭͱͯ͠ɺϖοτϘτϧ Λ౤ࣹޙɺ࠷ߴ఺ΑΓԼͰςʔϒϧʹୡͨ࣌͠ͷ଎౓͕࠷ খͱͳΔΑ͏ͳ৚݅ΛٻΊΔͷ΋Ұͭͷํ๏Ͱ͋Δɻ͜ͷ ৚݅ʹ͍ͭͯ͸ɺ์෺ӡಈʹ͓͚ΔมҐͷؔ܎͔ࣜΒɺ༩ ͑ΒΕͨ఺Λ௨ΔͨΊͷ౤ࣹॳ଎౓ͷؔ܎ࣜΛಋ͘͜ͱ͕ Ͱ͖Δɻ·ͨɺؔ࿈ͨ͠৘ใͱͯ͠์෺ӡಈʹ͓͚Δ࠷େ ͷਫฏ౸ୡڑ཭Λ༩͑Δڼ֯ʹ͍ͭͯͷใࠂ͕ͳ͞Ε͍ͯ Δɻ(1) ޙऀͷํ๏Λ༻͍Δ͜ͱͰ΋ɺϖοτϘτϧ͕ςʔ ϒϧͷ্ʹୡͨ͠ͱ͖ͷ଎౓Λ࠷খʹ͢Δ౤ࣹ৚͕݅ಘΒ ΕΔ͜ͱΛใࠂ͢Δɻ·ͣ͸ɺ࣍ͷΑ͏ʹ௨ৗͷ์෺ӡಈ ͔Βɺ౤ࣹ৚݅Λಋ͍ͯΈΔɻ ϦϕϥϧΞʔπܥ ˟861-1102 ۽ຊݝ߹ࢤࢢਢ԰ 2659-2

Faculty of Liberal Arts, 2659-2, Suya, Koshi-shi, Kumamoto, Japan 861-1102

∗∗ڌ఺ԽϓϩδΣΫτܥ

˟866-8501 ɹ۽ຊݝീ୅ࢢฏࢁ৽ொ 2627

Faculty of Project Centers, 2627 Hirayama-Shinmachi, Yatsushiro-shi, Kumamoto, Japan 866-8501

2. ์෺ӡಈͷيಓͷࣜΛ༻͍ͨಋग़ ౤ࣹҐஔΛݪ఺ʹͱΓɺॏྗՃ଎౓ͷେ͖͞Λgɺਫฏํ ޲Λxํ޲ɺԖ௚ํ޲Λyํ޲ͱ͢Δɻਤ1ͷ༷ʹॳ଎౓ ͷେ͖͞Λv0ɺڼ֯Λθͱͯ͠౤ࣹޙɺP఺(Ґஔ(X, h)) Λ௨ΔͱԾఆ͢Δɻ v0 x y X θ P(X, h) h y = tan θ · x − g 2v20 cos2 θ· x2 ਤ1 ์෺ӡಈ ͜ͷͱ͖ͷؔ܎ࣜ͸ɺ์෺ઢͷํఔࣜʹ୅ೖͯ͠ h = tan θ· X − gX 2 2v20cos2θ· · · (1) ͱͳΔɻ͜͜Ͱcos2θΛtan θΛ༻͍ͯॻ͖௚ͨࣜ͠͸ɺ gX2 2v20  1 + tan2θ− X · tan θ + h = 0· · · (2) ͱͳΔɻtan θ = qͱͯ͠ॻ͖௚͠ɺv02ʹ͍ͭͯղ͘ͱɺ v20= gX2 2 · 1 + q2 Xq− h· · · (3) ͱͳΔɻ͜͜ͰP఺Ͱͷ଎౓͕࠷΋খ͘͞ͳΔΑ͏ʹ͢Δ ۽ຊߴ౳ઐ໳ֶߍ ݚڀلཁɼୈ11 ߸ (2019) ― 98 ―

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์෺ӡಈʹ͓͚Δ࠷খͷ౸ୡ఺଎౓ͱ࠷େͷਫฏ౸ୡڑ཭ʹؔ͢Δߟ࡯ ʹ͸ॳ଎౓v0͕࠷খ͢ͳΘͪv02͕࠷খͱͳΕ͹Α͍ͷͰɺ (3)ࣜͰqͰඍ෼ͨࣜ͠Λ0ͱ͓͍ͯqʹ͍ͭͯղ͘ͱʢ2 ֊ඍ෼ͱۃখͷ৚͔݅Β΋ʣ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dv20 dq = gX2Xq2− 2hq − X 2 (Xq− h)2 = 0 d2v02 dq2 = gX2h2+ X2 (Xq− h)3 > 0 ˀ q= h + h2+ X2 X · · · (4) ͱͳΔɻ࠷খͱͳΔॳ଎౓vˆ0ΛٻΊΔͨΊ(4)ࣜΛ(3)ࣜ ʹ୅ೖͯ͠ ˆ v02= g  h + h2+ X2 ˀ vˆ0= g  h + h2+ X2 · · · (5) ͱٻΊΒΕΔɻ·ͨ৚݅Λຬͨ͢ڼ֯θˆʹ͍ͭͯͷؔ܎ࣜ ͸q = tan θΛʢ4)ࣜʹ୅ೖͯ͠ɺ tan ˆθ = h + h2+ X2 X · · · (6) ͱٻΊΒΕΔɻ࣍ʹɺ͜ͷؔ܎ࣜΛɺ࠷େͷਫฏ౸ୡڑ཭ Λ༩͑Δॳ଎౓ͷؔ܎͔Βಋ͍ͯΈΔɻ 3. ࠷େͷਫฏ౸ୡڑ཭Λ༩͑Δڼ֯Λ༻͍ͨಋग़ Ҏલͷใࠂ(1)Ͱ͸ɺ໰୊ͷఆٛΛਤ2ͷ༷ʹఆࣜԽ͠ɺ ౤ࣹ఺ͷߴ͞Λh,ॳ଎౓ͷେ͖͞Λv0ɼॏྗՃ଎౓Λg ͱͨ͠ͱ͖ͷམԼ఺·Ͱͷਫฏ౸ୡڑ཭͕࠷େ஋XˆͱͳΔ ڼ֯θˆΛٻΊͨɻ݁࿦͸ɺམԼ఺ͷ଎౓Λv1ͱ͢Δͱɺ v1= v20+ 2gh· · · (7) tan ˆθ = v0 v1 = v0 v20+ 2gh = 1 1 +2ghv2 0 · · · (8) ˆ X = v0· v1 g = v0 v20+ 2gh g · · · (9) Ͱ͋ͬͨɻ ࣍ʹɺਤ3ͷΑ͏ʹɺ౤ࣹ఺ΑΓམԼ఺͕h͚ͩߴ͍৔߹ Λߟ͑ΔɻҎԼɺམԼ఺Λ౸ୡ఺ͱݴ͍׵͑Δ͜ͱʹ͢Δɻ ͜ͷ৔߹ͷ଎౓ϕΫτϧਤ͸ਤ4ͷΑ͏ʹͳΔɻ౸ୡ఺ ͷਫฏڑ཭Xͱ౸ୡ఺·Ͱͷ࣌ؒt1ɺϕΫτϧਤͷ−→v0ͱ v 1Ͱ࡞ΒΕΔࡾ֯ܗͷ໘ੵSͱͷؒʹɺ X = v0cos θ· t1= g· t1· v0cos θ g = 2S g · · · (10) ͱ͍͏ؔ܎͕੒Γཱͭɻ X͕࠷େͷਫฏ౸ୡڑ཭Xˆ ͱͳΔͱ͖ͷ৚݅͸S͕࠷ େͱͳΔ৚݅Ͱ͋Δ͜ͱ͔Βਤ5ͷ༷ͳؔ܎͕੒Γཱͭɻ ͭ·Γɺॳ଎౓ͷେ͖͞ͱ౸ୡ఺ͷߴ͕ܾ͞·Ε͹ɺྗ ֶతΤωϧΪʔอଘଇͱɺॳ଎౓͓Αͼ౸ୡ఺ͷ଎౓ͷϕ θ v0 X h2 མԼ఺͕௿͍৔߹ θ →v0 X h m v1 ਤ3 མԼ఺͕ߴ͍৔߹ ൒ܘv1ͷԁ ൒ܘv0ͷԁ θ θˆ v0 v1 v1 v1 v0 v0 x y θ φ v0 v1 gt1 v0cos θ S = 1 2|−→v0| · |−→v1| sin φ =1 2· v0cos θ · g · t1 ਤ4 ౤ࣹ࣌ͱམԼ࣌ͷ଎౓ϕΫτϧ v1 ˆ θ v0 ˆ θ tan ˆθ =v0v15 ࠷େͷਫฏ౸ୡڑ཭ʹ͓͚Δ଎౓ϕΫτϧ

Research Reports of NIT, Kumamoto College. Vol.11 (2019)

― 99 ―

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Ϋτϧਤͷؔ܎Λ༻͍ͯ࠷େͷਫฏ౸ୡڑ཭Λදؔ͢܎͕ ಋ͔ΕΔɻΏ͑ʹɺ౤ࣹ఺ΑΓ౸ୡ఺͕h͚ͩߴ͍৔߹͸ɺ ࣜ(7),(8),(9)ͷhΛ−hʹஔ͖׵͑ͨ࣍ͷؔ܎ࣜ v1= v20− 2gh · · · ·(11) tan ˆθ = v0 v1 = 1 1−2ghv2 0 · · · ·(12) ˆ X = v0· v1 g = v0 v20− 2gh g · · · ·(13) ͕੒Γཱͭɻ ͯ͞ɺ͔͜͜Β͸౤ࣹ఺͔Βͷߴ͞h,ਫฏڑ཭X ͷ఺ ͕༩͑ΒΕͨ࣌ɺ౸ୡ఺Ͱͷ଎౓Λ࠷খʹ͢Δ৚݅Λಋ͍ ͯΈΔɻ౤ࣹ఺͔ΒਫฏมҐXɺߴ͞hͷ఺ʹ࠷খͷ଎౓ Ͱ౸ୡ͢ΔͨΊʹ͸ॳ଎౓v0ͷେ͖͞Λ࠷খʹͭͭ͠ɺ৚ ݅Λຬͨ͢ڼ֯ΛٻΊΔඞཁ͕͋Δɻ͜͜ͰX͕࠷େͷਫ ฏ౸ୡڑ཭ͱͳΔ৚݅Λຬͨͨ͠ॳ଎౓vˆ0ͱڼ֯θˆ͓Α ͼ౸ୡ଎౓vˆ1ͱͷؔ܎ΛٻΊͯΈΔɻ·ͣɺ ˆ v1= ˆ v02− 2gh · · · ·(14) tan ˆθ = vˆ0 ˆ v1 · · · ·(15) X = vˆ0· ˆv1 g = ˆ v0 vˆ02− 2gh g · · · ·(16) ͷؔ܎͕ຬͨ͞ΕΔͱͯ͠ɺࣜ(16)ΑΓɺ ˆ v02( ˆv02− 2gh) − g2X2= 0 · · · (17) ˀ vˆ02= g  h2+ X2 · · · (18) ͕ಋ͔ΕΔ͕ɺvˆ02> 0ΑΓ ˆ v02= g  h + h2+ X2 · · · (19) ˀ vˆ0= g  h + h2+ X2 · · · (20) ͱͳΔɻ͜ͷࣜ(19)Λࣜ(14),(15)ʹ୅ೖͯ͠ɺ ˆ v1= g  h + h2+ X2 − 2gh = g h2+ X2− h · · · ·(21) tan ˆθ = gh +√h2+ X2 g√h2+ X2− h =  h +√h2+ X22 X2 =h + h2+ X2 X · · · ·(22) ΛಘΔɻ͜ͷࣜ(20),(22)͸ɺઌʹٻΊͨɺࣜ(5),(6)ͱಉ ͡Ͱ͋Δɻ΋͜͠ͷ৚݅Λຬͨ͢v0ΑΓখ͍͞ॳ଎౓Ͱ͋ Ε͹౸ୡ఺ͷߴ͞hͷ࣌ͷ࠷େͷਫฏ౸ୡڑ཭͸XΑΓ খ͘͞ͳΔͷͰࢦఆ͞Εͨ఺ʹ౸ୡͰ͖ͳ͍ɻ͕ͨͬͯ͠ ্ͷ৚͕݅ɺ౸ୡͰ͖Δॳ଎౓ͷதͰ࠷খͷ΋ͷͰ͋Δͱ ݴ͑Δɻ ಋग़ͯ͠ΈΔͱɺͲͪΒͷํ๏Ͱ΋ൺֱత؆୯ʹٻΊΒ ΕΔ͜ͱ͕Θ͔͕ͬͨɺΉ͠Ζલઅͷํ๏ͷํ͕෼͔Γ΍ ͍͢ͷͰɺ࠷େͷਫฏ౸ୡڑ཭Λ༩͑Δڼ֯Λ࢖͏ํ๏ʹ ͸ϝϦοτ͸ݟ౰ͨΒͳ͍Α͏ʹ΋ࢥ͑Δɻ͔͠͠ɺ࣍અ Ͱड़΂ΔΑ͏ʹΉ͠Ζ྆ऀͷؔ܎͔Βٯʹ࠷େͷਫฏ౸ୡ ڑ཭Λ༩͑Δڼ֯Λલઅͷํ๏͔Βಋ͘͜ͱ͕Ͱ͖Δ͜ͱ ͕Θ͔Δɻ 4. ౸ୡ଎౓͕࠷খͷ৚݅ͱ౤ࣹํ޲ͷزԿֶతؔ܎ ʹ͍ͭͯ લઅ·ͰʹಘΒΕͨ౤ࣹ৚݅Λਤʹද͢ͱਤ6ͷΑ͏ʹ ͳΔɻ v1 v0 ˆ θ tan ˆθ =v0 v1 y =H Xx − 1+( HX )2 2H x2 P(X, h) x y H ɹ ਤ6 ࠷খ౸ୡ଎౓ͷ౤ࣹ৚݅ ͜ͷ࣌ͷ์෺ઢͷํఔࣜ͸,H = h +√h2+ X2ͱͯ͠ y = H X · x − 1 +HX2 2H · x 2· · · (23) ͱͳΔɻ͜ͷํఔࣜʹ͸ɺॏྗՃ଎౓g͕໌ࣔతʹ͸ؚ· Ε͍ͯͳ͍͕ɺ͜ͷيಓͱͳΔͨΊʹ͸ॳ଎౓ͷେ͖͞v0 ͕v0=√gHͰ͋Δ৚͕݅෇͘ͷͰɺॏྗՃ଎౓͕ؔΘͬ ͯ͘Δɻ͜ͷॳ଎౓͸ɺਅ্ʹ෺ମΛ౤্͛͛ͨͱ͖ɺH2 ͷߴ͞·Ͱ্͕Δ଎౓Ͱ͋Δɻ΋͠ਫฏڑ཭Xʹ੍ݶ͕ͳ ͍ͷͰ͋Ε͹౰વͷ͜ͱͳ͕ΒX = 0ͷ࣌ɺͭ·ΓP఺ ͷਅԼ͔Βv0 =√2ghͰ౤্͛͛ͨͱ͖ʹߴ͞hͷ౸ୡ ఺Ͱͷ଎౓v1= 0ͱͳΓ࠷௿ͷ౸ୡ఺଎౓ͱͳΔɻ ͱ͜ΖͰɺਤ6ʹ͓͍ͯɺॳ଎౓−→v0ͱ౸ୡ଎౓−→v1ͷ޲ ͖Λٯʹͯ͠ΈΔͱߴ͞hͷ఺͔Βͷ౤ࣹʹͳΔɻ౤ࣹӡ ಈͰ͸࣌ؒతʹ൓సͯ͠΋ಉ͡يಓΛඳ͘ͷͰ଎౓͕−−→v1 Ͱͭ·Γڼ֯π2 − ˆθͰ౤ࣹͨ͠ͱ͖ʹɺਫฏ౸ୡڑ཭͕࠷ େͷXͱͳΔɻ·ͨɺߴ͞h͔Β౤ࣹͯ͠ਫฏ౸ୡڑ཭ ۽ຊߴ౳ઐ໳ֶߍɹݚڀلཁɼୈ11 ߸ (2019) ᨺ≀㐠ື࡟࠾ࡅࡿ᭱ᑠࡢ฿㐩Ⅼ㏿ᗘ࡜᭱኱ࡢỈᖹ฿㐩㊥㞳࡟㛵ࡍࡿ⪃ᐹ ― 100 ―

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์෺ӡಈʹ͓͚Δ࠷খͷ౸ୡ఺଎౓ͱ࠷େͷਫฏ౸ୡڑ཭ʹؔ͢Δߟ࡯ v1 v0 ˆ θ tan ˆθ =v0v1 y =X Hx −1+( XH ) 2 2X x2 P(X, −h) x y H −h H =√h2+ X2 O ਤ7 མԼ఺͕௿͍৔߹ͷ࠷େ౸ୡڑ཭ͱ౤ࣹ ৚݅ Xͷ఺ʹୡͨ͠ͱ͖ͷ଎౓͕࠷খʹͳΔ৚݅ͱͯ͠લઅͱ ಉ༷ͷٞ࿦͕Ͱ͖Δ͜ͱ͕Θ͔Δɻ܁Γฦ͠ʹͳΔ͕ɺ̎ ͭͷ໰୊ େ͖͕͞Ұఆͷॳ଎౓Ͱɺߴ͞hͷ఺͔Β౤ࣹͨ͠ͱ ͖ͷਫฏ౸ୡڑ཭Λ࠷େʹ͢Δॳ଎౓ͷ޲͖(ڼ֯)Λ ܾఆ͢Δɻ ߴ͞hͰਫฏ౸ୡڑ཭X ͷ఺ʹ౸ୡ͢ΔͨΊͷ࠷খ ͷॳ଎౓ͷେ͖͞ͱ޲͖ʢڼ֯ʣΛܾఆ͢Δɻ ͸ɺಉؔ͡܎ࣜΛݩʹղܾ͞ΕΔͱݴ͑Δɻ͜ͷ͜ͱΑ Γٯʹࣜ(5),(6)͔Β X = vˆ0 2 g 1−2gh ˆ v02 · · · ·(24) tan ˆθ = 1 1−2ghvˆ 02 · · · ·(25) ͕ಋग़͞ΕΔɻߋʹࣜ(25)ͰhΛ−hʹɺvˆ0Λv0ʹஔ͖ ׵͑Ε͹ࣜ(8)ͱಉ͡ͱͳΔɻ͕ͨͬͯ͠ɺࢦఆ͞Εͨ఺ Λ࠷খͷ౸ୡ଎౓Ͱ௨ΔΑ͏ͳॳ଎౓ͱڼ֯ΛٻΊΔؔ܎ ͔ࣜΒɺ౤ࣹ఺ͱ౸ୡ఺ͷߴ͞ͷࠩͱॳ଎౓͔Β࠷େͷਫ ฏ౸ୡڑ཭͓Αͼॳ଎౓ͷڼ֯Λදؔ͢܎͕ࣜٻΊΒΕΔɻ ਤ2ͷঢ়گΛਤ6Λݩʹਤ7ͷ༷ʹม͑ͯඳ͍ͯΈΔɻ͜ ΕʹΑΓߴ͞hͷ఺͔Β౤ࣹ͢Δͱ͍͏ঢ়گΛ౤ࣹ఺͔Β h͚ͩ௿͍ͱ͜ΖʹΉ͚ͯ౤ࣹ͢ΔͱมΘΔɻ͜ΕʹΑΓ ॳ଎౓ͷେ͖͞v0͕༩͑ΒΕͨ࣌ɺ࠷େ౸ୡڑ཭ͱͯ͠ XΛ(9)ࣜͷӈลΛ༻͍ͯٻΊɺͦ͜ʹ౸ୡ͢ΔΑ͏ͳॳ ଎౓ͷڼ֯ͱͯ͠ਤ7ͷΑ͏ͳزԿֶతͳঢ়گ͕ܾఆ͞Ε Δࣄ͕Θ͔Δɻ g, hΛఆ਺ͱͯ͠ɺ3ͭͷม਺v0, X, qͷؒʹ(3)͕ࣜ ੒Γཱͭͱ͖ɺӡಈͱ࣮ͯ͠ݱ͞ΕΔൣғͰɺ ∂X ∂q = 0⇐⇒ ∂v0 ∂q = 0· · · (26) ͕੒ΓཱͪɺҎ্ͷΑ͏ͳؔ܎͕ಋ͔ΕΔͱ͍͑Δɻ(3)ࣜ ͷΠϝʔδΛάϥϑʹද͢ͱਤ8ͷ༷ʹͳΔɻ ਤ8 ॳ଎౓ͷ2৐(v20)ͱਫฏ౸ୡڑ཭X, ڼ ֯ͷਖ਼઀qͷؔ܎ ্Լ͕࣠2v20 g Λද͠ɺଞͷ͕࣠XͱqΛද͍ͯ͠Δɻh ͸1ͱͯ͠ඳ͍ͯ͋Δɻਤ8ΑΓɺv0͕ҰఆͰɺX͕࠷େ ͱͳΔqΛ݁ͿۂઢͱɺX͕ҰఆͰv0͕࠷খͱͳΔqΛ ݁Ϳۂઢ͕Ұக͢Δঢ়گ͕ΠϝʔδͰ͖Δɻ 5. Ή ͢ ͼ ॳ౳෺ཧͷֶश՝୊Ͱ͋Δ์෺ӡಈͷԠ༻ͱͯ͠ɺ౸ୡ ఺ͷ଎౓Λ࠷খʹ͢ΔΑ͏ͳॳ଎౓ͷ໰୊Λఏࣔ͠ɺղΛ 2௨Γͷํ๏Ͱࣔ͠ɺͦͷؔ܎ʹ͍ͭͯߟ࡯ͨ͠ɻ ۽ຊߴઐ۽ຊΩϟϯύεͷϩϘίϯ෦ͷֶੜʹ͸2018೥ ౓ͷίϯςετʹ͓͚ΔϩϘοτ੡࡞࣌ʹ૬ஊΛड͚ͨͷ Ͱɺࠓճࣔͨ͠࠷େ౸ୡڑ཭ͷڼ֯Λ༻͍ͨํ๏Ͱ৚݅Λ ಋ͖ɺղΛ͕ࣔͨ͠ɺ࣮ࡍͷϩϘοτͭ͘Γʹ͸࢖ΘΕͳ ͔ͬͨΑ͏Ͱ͋ΔɻϖοτϘτϧΛ҆ఆͯ͠ඈ͹ٕ͢ज़΍ ண஍࣌ͷিܸͰ౗Εͳ͍Α͏ʹ͢Δٕज़ͳͲ༷ʑͳ͜ͱΛ ߟ͑Δඞཁ͕͋Γɺ୯ͳΔ࣭఺ͷӡಈ͚ͩͰ͸ࢀߟʹͳΒ ͳ͔ͬͨΑ͏ͩɻ࣮ࡍͷίϯςετͷϧʔϧͰ͸ɺࣗಈϩ Ϙοτ͸ΰʔϧͷςʔϒϧʹ͍͍ۙͮͯ͘͜ͱ͕Ͱ͖Δͨ ΊɺϖοτϘτϧͷண஍ͷিܸΛখ͘͢͞Δʹ͸Ͱ͖Δͩ ͚ΰʔϧͷ͔ۙ͘Β౤ࣹ͢Δํ͕࠷దͳ৚݅Λಘ΍͍͢ɻ ʢྩ࿨ݩ೥9݄25೔ड෇ʣ ʢྩ࿨ݩ೥12݄5೔डཧʣ ࢀߟจݙ ʢ1 ʣ ޻౻༑༟ɹଞɼʠ ์෺ӡಈͷ࠷େ౸ୡڑ཭Λ༩͑Δڼ֯Λ؆୯ʹಋ͘ ํ๏ ʡ ɼ۽ຊߴ౳ઐ໳ֶߍݚڀلཁ ୈ5 ߸, pp.116-119ɽʢ2013ʣ

Research Reports of NIT, Kumamoto College. Vol.11 (2019)

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