Wiki source code of 2025-Ryudai-Butsuri-Q2A(3)ver02

Last modified by Akiyoshi Yamakawa on 2026/02/26 20:36
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 1: = 琉大2025物理大問２A問3の計算式 =
 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)**
 4: 
 5: * **最初の落下までの時間をｔ,,0,,とおき、
 6: 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、
 7: 床に衝突する直前の鉛直方向速さをV,,y　,,とおくと、
 8: 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**
 9: \\(% style="font-size: 20px; font-family: Georgia, serif" %)**//(//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//1/2//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//)g//**(% style="font-family: Georgia, serif; font-size: 22px" %)**//t//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//,,0,,^^2^^=8h//**(% style="font-size:20px;" %)**　より　**(% style="font-family: Georgia, serif; font-size: 22px" %)**//t//**(% style="font-family: Georgia, serif; font-size: 20px" %)**//,,0,, = 4√(h/g)//**(%%)
10: (% style="font-size:18px;" %)** **(%%)
11: (% style="font-family: Georgia, serif; font-size: 22px" %)**//vt//**(% style="font-family: Georgia, serif; font-size: 20px" %)**// = L//**(% style="font-size:20px;" %)** より、**(% style="font-family: Georgia, serif; font-size: 22px" %)**//t//**(% style="font-family: Georgia, serif; font-size: 20px" %)**// //= //L/v//**(%%)
12: (% style="font-size:18px;" %)** **(%%)
13: (% style="font-family: Georgia, serif; font-size: 20px" %)**//(//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//1/2//**(% style="font-family: Georgia, serif; font-size: 20px" %)**//)m//**(% style="font-family: Georgia, serif; font-size: 22px" %)**//v//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//,,y,,^^2^^=mh (8h)//**(% style="font-size:20px;" %)**// //より、**(% style="font-size: 20px; font-family: Georgia, serif" %)**//u=(//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//1/2//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//)//**(% style="font-family: Georgia, serif; font-size: 22px" %)**//v//**(% style="font-family: Georgia, serif; font-size: 20px" %)**//,,y,,=//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//2//**(% style="font-family: Georgia, serif; font-size: 20px" %)**//√//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//(gh)//**(%%)
14:  
15: * **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、
16: \\(% style="font-family: Georgia, serif; font-size: 22px" %)//h < y=u (t - t,,0,,) - (//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 22px" %)//1/2//(% style="font-family: Georgia, serif; font-size: 22px" %)//)g(t - t,,0,,)^^2^^//(%%)**
17: \\**これに、上記のu、t、t,,0,,を代入すると、**
18: \\(% style="font-size:22px;" %)**h<2**(% style="font-size: 20px; font-family: Georgia, serif" %)**//√(gh)(L/v - //**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//4//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//√(hg)) - (//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//1/2//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//)g(L/v-//**(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)**//4//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//√(hg))^^2^^//**(%%)
19: \\(% style="font-size:22px;" %)**//h<2//**(% style="font-size: 22px; font-family: Georgia, serif" %)**//(L/v)√(gh) - 8h - //**(% style="font-size:22px;" %)**//(1/2//**(% style="font-size: 20px; font-family: Georgia, serif; font-size: 22px" %)**//)g(L^^2^^/v^^2^^) - 8h + 4L/v√(gh)//**(%%)
20: (% style="font-size: 20px; font-family: Georgia, serif" %)**// //**(%%)
21: (% style="font-size:22px;" %)**//h<6//**(% style="font-size: 22px; font-family: Georgia, serif" %)**//(L/v)√(gh) //**(% style="font-size: 20px; font-family: Georgia, serif" %)**//- //**(% style="font-size:22px;" %)**//17//**(% style="font-size: 20px; font-family: Georgia, serif" %)**//h - //**(% style="font-size:22px;" %)**//(1/2//**(% style="font-size: 20px; font-family: Georgia, serif; font-size: 22px" %)**//)g(L^^2^^/v^^2^^) //**(%%)**これを、ｖの二次不等式に直すと
22: (% style="font-size: 20px; font-family: Georgia, serif; font-size: 22px" %)//　//(%%)
23: (% style="font-family: Georgia, serif; font-size: 22px" %)//０>//(% style="font-size:22px;" %)//34//(% style="font-family: Georgia, serif; font-size: 22px" %)//v^^2^^ - //(% style="font-size:22px;" %)//12//(% style="font-family: Georgia, serif; font-size: 22px" %)//L√(g/h)v //(% style="font-family: Georgia, serif; font-size: 20px" %)//+  //(% style="font-size: 20px; font-family: Georgia, serif; font-size: 22px" %)//(g/h)L^^2^^  //(%%)さらに、＝０の二次方程式としてvを求めると、
24: 　
25: (% style="font-size: 20px; font-family: Georgia, serif" %)//v=[//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6//(% style="font-size: 20px; font-family: Georgia, serif" %)//√(g/h)±√{//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//36//(% style="font-size: 20px; font-family: Georgia, serif" %)//(g/h)L^^2^^ - //(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//34//(% style="font-size: 20px; font-family: Georgia, serif" %)//(g/h)L^^2^^}]///(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//34//(% style="font-size: 20px; font-family: Georgia, serif" %)// =[(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6//(% style="font-size: 20px; font-family: Georgia, serif" %)//±√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)L√(g/h)]///(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//34//(% style="font-size: 20px; font-family: Georgia, serif" %)// //(%%)ｖの下限なので、－の方をとって、
26: \\(% style="font-size: 20px; font-family: Georgia, serif" %)//v=[(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 - //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)L√(g/h)]///(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//34//(% style="font-size: 20px; font-family: Georgia, serif" %)// = [(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 - //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 + //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)L√(g/h)]///(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//{34//(% style="font-size: 20px; font-family: Georgia, serif" %)//(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 + //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)}=[34L√(g/h)]///(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//{34//(% style="font-size: 20px; font-family: Georgia, serif" %)//(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 + //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)}
27: \\ =[L√(g/h)]/(//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//6 + //(% style="font-size: 20px; font-family: Georgia, serif" %)//√//(% style="font-family: Arial, Helvetica, sans-serif; font-size: 20px" %)//2//(% style="font-size: 20px; font-family: Georgia, serif" %)//)　//(% style="font-size: 16px; font-family: Arial, Helvetica, sans-serif" %)・・・答え(%%)**
28: **（注：次元解析して速度の単位になっていることも確認！　m(% style="font-family:Georgia,serif;" %)√(%%){(m/s^^2^^)/m}=m/s )**