Wiki source code of 2025-Ryudai-Butsuri-Q2A(3)-ver03

Last modified by Akiyoshi Yamakawa on 2026/02/26 20:31
= 琉大2025物理大問２A問3の計算式 =

(% 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)**

* (% 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、
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、
 **
** 　[[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"]]
 
** 　[[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"]]
 
** 　[[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"]]
 
* (% 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、**(%%)**
 **
** [[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、**(%%)
\\[[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"]]
\\[[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"]]
\\[[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、(%%)**
\\[[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、**(%%)
\\[[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、**(%%)
\\[[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**(%%)
\\**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"]]