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문제

스키 점프는 가장 인기있는 겨울 스포츠 대회 중 하나이다. 스키 점프 언덕은 기록을 높이기 위해 점점 커졌다. 선수들의 안전을 위해서 국제 스키 연맹은 착륙 속도와 각도의 최대값을 정해두었다. 새로 만들어진 스키 점프 경기장이 주어진다. 출발대의 가장 높은 곳에서 출발했을 때, 착륙 속도와 각도를 구하는 프로그램을 작성하시오.

이 문제를 풀기 위해서는 아래와 같은 간단한 물리 지식이 필요하다. 

언덕은 다음과 같은 모양이다.

l은 언덕이 시작하는 곳을 원점으로 했을 때, x축 상의 위치이다. H는 언덕의 높이, L은 언덕의 너비이다. j는 스키 점프를 시작할 수 있는 최대 높이이고, p는 점프대와 언덕 꼭대기와의 높이 차이이다. 마찰력과 공기저항은 무시한다면, 선수는 다음과 같은 곡선을 그리며 날아가게 된다.

여기서 v0은 점프할 때의 속도이다. 이 속도는 에너지 보존의 법칙을 이용해서 계산할 수 있다. 포텐셜 에너지와 운동 에너지의 정의는 다음과 같다.

모든 식에서 g는 중력 상수이며, 9.81ms-2로 계산한다.

입력

첫째 줄에 테스트 케이스의 개수 t가 주어진다. (0 < t < 160,000)

각 테스트 케이스는 한 줄로 이루어져 있고, j, p, H, L이 주어진다. (0 < j, p, H, L ≤ 500) 모든 값의 단위는 미터이다.

출력

각 테스트 케이스에 대해서, 다음과 같은 세 가지 값을 출력한다.

1. x축 상에서 착륙 위치 l

2. 착륙 속도 |vl| (m/s)

3. 언덕에 대한 각도 α (degree)

오차는 10-4까지 허용한다.

예제 입력 1

3
50 5 10 100
50 5 30 100
50 5 50 100

예제 출력 1

40.82482905 33.83045965 12.93315449
81.04978134 40.31656580 26.21334827
104.8808848 45.38832449 46.36470132

힌트

두 벡터 a와 b의 내적은 다음과 같이 구할 수 있다.

[{"problem_id":"3959","problem_lang":"0","title":"\uc2a4\ud0a4 \uc810\ud504","description":"<p>\uc2a4\ud0a4 \uc810\ud504\ub294 \uac00\uc7a5 \uc778\uae30\uc788\ub294 \uaca8\uc6b8 \uc2a4\ud3ec\uce20 \ub300\ud68c \uc911 \ud558\ub098\uc774\ub2e4. \uc2a4\ud0a4 \uc810\ud504 \uc5b8\ub355\uc740 \uae30\ub85d\uc744 \ub192\uc774\uae30 \uc704\ud574 \uc810\uc810 \ucee4\uc84c\ub2e4. \uc120\uc218\ub4e4\uc758 \uc548\uc804\uc744 \uc704\ud574\uc11c \uad6d\uc81c \uc2a4\ud0a4 \uc5f0\ub9f9\uc740 \ucc29\ub959 \uc18d\ub3c4\uc640 \uac01\ub3c4\uc758 \ucd5c\ub300\uac12\uc744 \uc815\ud574\ub450\uc5c8\ub2e4. \uc0c8\ub85c \ub9cc\ub4e4\uc5b4\uc9c4 \uc2a4\ud0a4 \uc810\ud504 \uacbd\uae30\uc7a5\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ucd9c\ubc1c\ub300\uc758 \uac00\uc7a5 \ub192\uc740 \uacf3\uc5d0\uc11c \ucd9c\ubc1c\ud588\uc744 \ub54c, \ucc29\ub959 \uc18d\ub3c4\uc640 \uac01\ub3c4\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n\r\n\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj1.png\" style=\"height:363px; width:593px\" \/><\/p>\r\n\r\n\r\n\r\n<p>\uc774 \ubb38\uc81c\ub97c \ud480\uae30 \uc704\ud574\uc11c\ub294 \uc544\ub798\uc640 \uac19\uc740 \uac04\ub2e8\ud55c \ubb3c\ub9ac \uc9c0\uc2dd\uc774 \ud544\uc694\ud558\ub2e4.&nbsp;<\/p>\r\n\r\n\r\n\r\n<p>\uc5b8\ub355\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ubaa8\uc591\uc774\ub2e4.<\/p>\r\n\r\n\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj2.png\" style=\"height:104px; width:429px\" \/><\/p>\r\n\r\n\r\n\r\n<p>l\uc740 \uc5b8\ub355\uc774 \uc2dc\uc791\ud558\ub294 \uacf3\uc744 \uc6d0\uc810\uc73c\ub85c \ud588\uc744 \ub54c, x\ucd95 \uc0c1\uc758 \uc704\uce58\uc774\ub2e4. H\ub294 \uc5b8\ub355\uc758 \ub192\uc774, L\uc740 \uc5b8\ub355\uc758 \ub108\ube44\uc774\ub2e4. j\ub294 \uc2a4\ud0a4 \uc810\ud504\ub97c \uc2dc\uc791\ud560 \uc218 \uc788\ub294 \ucd5c\ub300 \ub192\uc774\uc774\uace0, p\ub294 \uc810\ud504\ub300\uc640 \uc5b8\ub355 \uaf2d\ub300\uae30\uc640\uc758 \ub192\uc774 \ucc28\uc774\uc774\ub2e4. \ub9c8\ucc30\ub825\uacfc \uacf5\uae30\uc800\ud56d\uc740 \ubb34\uc2dc\ud55c\ub2e4\uba74, \uc120\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uace1\uc120\uc744 \uadf8\ub9ac\uba70 \ub0a0\uc544\uac00\uac8c \ub41c\ub2e4.<\/p>\r\n\r\n\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj3.png\" style=\"height:63px; width:489px\" \/><\/p>\r\n\r\n\r\n\r\n<p>\uc5ec\uae30\uc11c v0\uc740 \uc810\ud504\ud560 \ub54c\uc758 \uc18d\ub3c4\uc774\ub2e4. \uc774 \uc18d\ub3c4\ub294 \uc5d0\ub108\uc9c0 \ubcf4\uc874\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud574\uc11c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \ud3ec\ud150\uc15c \uc5d0\ub108\uc9c0\uc640 \uc6b4\ub3d9 \uc5d0\ub108\uc9c0\uc758 \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj4.png\" style=\"height:52px; width:495px\" \/><\/p>\r\n\r\n\r\n\r\n<p>\ubaa8\ub4e0 \uc2dd\uc5d0\uc11c g\ub294 \uc911\ub825 \uc0c1\uc218\uc774\uba70, 9.81ms<sup>-2<\/sup>\ub85c \uacc4\uc0b0\ud55c\ub2e4.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 t\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; t &lt; 160,000)<\/p>\r\n\r\n\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ud55c \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, j, p, H, L\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; j, p, H, L &le; 500) \ubaa8\ub4e0 \uac12\uc758 \ub2e8\uc704\ub294 \ubbf8\ud130\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \uac00\uc9c0 \uac12\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n\r\n\r\n<p>1. x\ucd95 \uc0c1\uc5d0\uc11c \ucc29\ub959 \uc704\uce58 l<\/p>\r\n\r\n\r\n\r\n<p>2. \ucc29\ub959 \uc18d\ub3c4 |vl| (m\/s)<\/p>\r\n\r\n\r\n\r\n<p>3. \uc5b8\ub355\uc5d0 \ub300\ud55c \uac01\ub3c4 &alpha; (degree)<\/p>\r\n\r\n\r\n\r\n<p>\uc624\ucc28\ub294 10<sup>-4<\/sup>\uae4c\uc9c0 \ud5c8\uc6a9\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\ub450 \ubca1\ud130 a\uc640 b\uc758 \ub0b4\uc801\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj5.png\" style=\"height:37px; width:388px\" \/><\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3959","problem_lang":"1","title":"Ski Jumping","description":"<p>Ski jumping is one of the most popular winter sport competitions. In the chase of records, ski jumping hills become larger and larger. To ensure the safety of the competitors, landing speed and angle must not exceed critical margins de ned by the FIS. Today, it&#39;s your task to assess these values for a newly constructed ski jumping arena shown in the gure.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj1.png\" style=\"height:363px; width:593px\" \/><\/p>\r\n\r\n<p>Instead of doing measurements in the eld, you can use a little math to solve your problem, since the hill has the following shape:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj2.png\" style=\"height:104px; width:429px\" \/><\/p>\r\n\r\n<p>where l is the position on the x-axis with its origin in the beginning of the hill. H is the height and L is the width of the hill; j is the maximum starting height of the ski-jump and p is the height difference between the end of the (ski-jump) approach and the top of the hill. Assuming that friction plays no important role and since the critical margins are defi ned for a flight without any influence of wind, you may utilize the following flight curve:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj3.png\" style=\"height:63px; width:489px\" \/><\/p>\r\n\r\n<p>where v0 is the speed gained in the approach. You can obtain this value from the law of energy conservation. Potential and kinetic energy are de ned as follows:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj4.png\" style=\"height:52px; width:495px\" \/><\/p>\r\n\r\n<p>In all equations, g is the gravitational constant (g &asymp; 9:81 ms<sup>-2<\/sup>)<\/p>\r\n","input":"<p>Input starts with the number of test cases t on a single line (0 &lt; t &lt; 160 000).<\/p>\r\n\r\n<p>Every test case consists of a single line containing four positive integers j, p, H, and L as de fined in the problem statement (0 &lt; j, p, H, L &le;\u0014 500). The unit of all values is meter.<\/p>\r\n","output":"<p>For every test case, print one line containing<\/p>\r\n\r\n<ul>\r\n\t<li>the landing position l on the x-axis,<\/li>\r\n\t<li>the landing speed |v<sub>l<\/sub>| of the jumper (in meters per second), and<\/li>\r\n\t<li>the speed-angle &alpha; (in degree) with respect to the hill (see the figure).<\/li>\r\n<\/ul>\r\n\r\n<p>The values must be separated by a single blank. An absolute or relative error of 10<sup>-4<\/sup> is tolerated<\/p>\r\n","hint":"<p>The inner product of two vectors a and b is de fined as:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sj5.png\" style=\"height:37px; width:388px\" \/><\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]