Changes for page 2025-Ryudai-Butsuri-Q2A(3)-ver03
Last modified by Akiyoshi Yamakawa on 2026/02/26 20:31
From version 11.1
edited by Akiyoshi Yamakawa
on 2026/02/25 20:53
on 2026/02/25 20:53
To version 12.1
edited by Akiyoshi Yamakawa
on 2026/02/26 20:31
on 2026/02/26 20:31
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... ... @@ -1,9 +1,10 @@ 1 1 = 琉大2025物理大問２A問3の計算式 = 2 2 3 -**世間に、計算途中式の資料が見当たらないので、途中式を書いてみました。(by A.YK)** 3 +(% style="font-size:18px;" %)**世間に、計算途中式の資料が見当たらないので、途中式を書いてみました。(by A.YK)** 4 4 5 -* **最初の落下までの時間をｔ,,0,,とおき、高さｈの障害物の位置まで移動するのにかかる時間をｔ、 6 -床に衝突する直前の鉛直方向速さをV,,y ,,とおくと、反発係数e=1/2より、衝突直後の鉛直方向速さｕ=1/2V,,y,,であり、** 5 +* (% style="font-size:18px;" %)**最初の落下までの時間をｔ,,0,,とおき、高さｈの障害物の位置まで移動するのにかかる時間をｔ、 6 +床に衝突する直前の鉛直方向速さをV,,y ,,とおくと、反発係数e=1/2より、衝突直後の鉛直方向速さｕ=1/2V,,y,,であり、 7 + ** 7 7 ** [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{1}{2}g{t_0}^2=8h]] より、[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?{t_0}=4\sqrt{\frac{h}{g}}||height="33" width="83"]] 8 8 9 9 ** [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?vt=L]] より、[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?t=||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?vt=L"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?vt=L" height="30" width="19"]] ... ... @@ -10,13 +10,13 @@ 10 10 11 11 ** [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{1}{2}m{v_y}^2=mg8h]] より、[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?u=\frac{1}{2}v_y=2\sqrt{gh}||height="26" width="130"]] 12 12 13 -* **時刻ｔでの高さｙが障害物の高さｈよりも上であればよいので、 14 +* (% style="font-size:18px;" %)**時刻ｔでの高さｙが障害物の高さｈよりも上であればよいので、**(%%)** 14 14 ** 15 -** [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<u(t-t_0)-\frac{1}{2}g(t-t_0)^2]] **これに、上記のｕ、ｔ、ｔ,,0,,を代入すると、** 16 +** [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<u(t-t_0)-\frac{1}{2}g(t-t_0)^2]] (% style="font-size:18px;" %)**これに、上記のｕ、ｔ、ｔ,,0,,を代入すると、**(%%) 16 16 \\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="20" width="76"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?-4\sqrt{\frac{h}{g}})-\frac{1}{2}g(||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="35" width="123"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?-4\sqrt{\frac{h}{g}})^2||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="35" width="79"]] 17 17 \\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\frac{L}{v}\sqrt{gh}-8h-\frac{1}{2}g\frac{L^2}{v^2}-8h+4\frac{L}{v}\sqrt{gh}" height="14" width="35"]] [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]] [[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\sqrt{gh}-8h-\frac{1}{2}g\frac{L^2}{v^2}-8h+4||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\frac{L}{v}\sqrt{gh}-8h-\frac{1}{2}g\frac{L^2}{v^2}-8h+4\frac{L}{v}\sqrt{gh}" height="42" width="250"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\sqrt{gh}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\frac{L}{v}\sqrt{gh}-8h-\frac{1}{2}g\frac{L^2}{v^2}-8h+4\frac{L}{v}\sqrt{gh}" height="21" width="40"]] 18 -\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}\frac{L}{v}-17h-\frac{1}{2}g\frac{L^2}{v^2}" height="24" width="90"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?-17h-\frac{1}{2}g\frac{L^2}{v^2}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}\frac{L}{v}-17h-\frac{1}{2}g\frac{L^2}{v^2}" height="42" width="130"]]** これを、ｖの二次不等式に直すと、** 19 -\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?0>34v^2-12\sqrt{\frac{g}{h}}Lv+\frac{g}{h}L^2||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0>34v^2-12\sqrt{gh}v+\frac{g}{h}" height="40" width="200"]] **さらに、＝０の二次方程式としてvを求めると、** 20 -\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?v={\frac{6\sqrt{\frac{g}{h}}L\pm\sqrt{36\frac{g}{h}L^2-34\frac{g}{h}L^2}}{34}=\frac{(6\pm\sqrt{2})\sqrt{\frac{g}{h}}L}{34}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{6\frac{g}{h}\pm\sqrt{36\frac{g}{h}-34\frac{g}{h}}}{34}=\frac{(6\pm\sqrt{2})\sqrt{\frac{g}{h}}}{34}" height="65" width="474"]] **ｖの下限なので、－の方をとって、** 21 -\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{(6-sqrt{2})}{34}\sqrt{\frac{g}{h}}L=\frac{(6-sqrt{2})(6+sqrt{2})}{34(6+sqrt{2})}\sqrt{\frac{g}{h}}L=\frac{34}{34(6+\sqrt{2})}\sqrt{\frac{g}{h}}L=\frac{1}{(6+\sqrt{2})}\sqrt{\frac{g}{h}}L||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{(6-sqrt{2})}{34}\sqrt{\frac{g}{h}}}=\frac{(6-sqrt{2})(6+sqrt{2})}{34(6+sqrt{2})}}\sqrt{\frac{g}{h}}=\frac{34}{34(6+\sqrt{2})}\sqrt{\frac{g}{h}}=\frac{1}{(6+\sqrt{2})}\sqrt{\frac{g}{h}}" height="55" width="747"]] 22 -\\**注：次元解析して速度の単位になっていることも確認！ (% style="font-size: 18px; font-family: Georgia,serif" %)√(%style="font-size:18px;" %){(m/s^^2^^)/m)}m=m/s(%%)**19 +\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}\frac{L}{v}-17h-\frac{1}{2}g\frac{L^2}{v^2}" height="24" width="90"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\frac{L}{v}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?h<2\sqrt{gh}(\frac{L}{v}-4\sqrt{hg})-\frac{1}{2}g(\frac{L}{v}-4\sqrt{hg})^2" height="31" width="20"]][[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?-17h-\frac{1}{2}g\frac{L^2}{v^2}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0<6\sqrt{gh}\frac{L}{v}-17h-\frac{1}{2}g\frac{L^2}{v^2}" height="42" width="130"]]** (% style="font-size:18px;" %)これを、ｖの二次不等式に直すと、(%%)** 20 +\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?0>34v^2-12\sqrt{\frac{g}{h}}Lv+\frac{g}{h}L^2||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?0>34v^2-12\sqrt{gh}v+\frac{g}{h}" height="40" width="200"]] (% style="font-size:18px;" %)**さらに、＝０の二次方程式としてvを求めると、**(%%) 21 +\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?v={\frac{6\sqrt{\frac{g}{h}}L\pm\sqrt{36\frac{g}{h}L^2-34\frac{g}{h}L^2}}{34}=\frac{(6\pm\sqrt{2})\sqrt{\frac{g}{h}}L}{34}||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{6\frac{g}{h}\pm\sqrt{36\frac{g}{h}-34\frac{g}{h}}}{34}=\frac{(6\pm\sqrt{2})\sqrt{\frac{g}{h}}}{34}" height="65" width="474"]] (% style="font-size:18px;" %)**ｖの下限なので、－の方をとって、**(%%) 22 +\\[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{(6-sqrt{2})}{34}\sqrt{\frac{g}{h}}L=\frac{(6-sqrt{2})(6+sqrt{2})}{34(6+sqrt{2})}\sqrt{\frac{g}{h}}L=\frac{34}{34(6+\sqrt{2})}\sqrt{\frac{g}{h}}L=\frac{1}{(6+\sqrt{2})}\sqrt{\frac{g}{h}}L||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?v=\frac{(6-sqrt{2})}{34}\sqrt{\frac{g}{h}}}=\frac{(6-sqrt{2})(6+sqrt{2})}{34(6+sqrt{2})}}\sqrt{\frac{g}{h}}=\frac{34}{34(6+\sqrt{2})}\sqrt{\frac{g}{h}}=\frac{1}{(6+\sqrt{2})}\sqrt{\frac{g}{h}}" height="55" width="747"]](% style="font-size:18px;" %)**・・・答え**(%%) 23 +\\**注：次元解析して速度の単位になっていることも確認！ **[[image:http://www.dietpanda.com/cgi-bin/mimetex.cgi?\sqrt{\frac{g}{h}}L{\rightarrow}\sqrt{\frac{m/s^2}{m}}{\cdot}m=m/s||alt="http://www.dietpanda.com/cgi-bin/mimetex.cgi?{t_0}=4\sqrt{\frac{h}{g}}" height="40" width="224"]]