2016 Multi-University Training Contest 3 Rower Bo

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Rower Bo

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 867    Accepted Submission(s): 296
Special Judge

Problem Description

There is a river on the Cartesian coordinate system,the river is flowing along the x-axis direction.

Rower Bo is placed at  (0,a) at first.He wants to get to origin  (0,0) by boat.Boat speed relative to water is  v1,and the speed of the water flow is  v2.He will adjust the direction of  v1 to origin all the time.

Your task is to calculate how much time he will use to get to origin.Your answer should be rounded to four decimal places.

If he can't arrive origin anyway,print"Infinity"(without quotation marks).

 

Input

There are several test cases. (no more than 1000)

For each test case,there is only one line containing three integers  a,v1,v2.

0≤a≤100,  0≤v1,v2,≤100,  a,v1,v2 are integers

 

Output

For each test case,print a string or a real number.

If the absolute error between your answer and the standard answer is no more than  10−4, your solution will be accepted.

 

Sample Input

2 3 3 2 4 3

 

Sample Output

Infinity 1.1428571429

 

Source

2016 Multi-University Training Contest 3

首先这个题微分方程强解显然是可以的，但是可以发现如果设参比较巧妙就能得到很方便的做法。

先分解v_1v​1​​，

设船到原点的距离是$r$，容易列出方程

\frac{ dr}{ dt}=v_2\cos \theta-v_1​dt​​dr​​=v​2​​cosθ−v​1​​

\frac{ dx}{ dt}=v_2-v_1\cos \theta​dt​​dx​​=v​2​​−v​1​​cosθ

上下界都是清晰的，定积分一下：

0-a=v_2\int_0^T\cos\theta{ d}t-v_1T0−a=v​2​​∫​0​T​​cosθdt−v​1​​T

0-0=v_2T-v_1\int_0^T\cos\theta{ d}t0−0=v​2​​T−v​1​​∫​0​T​​cosθdt

直接把第一个式子代到第二个里面

v_2T=\frac{v_1}{v_2}(-a+v_1T)v​2​​T=​v​2​​​​v​1​​​​(−a+v​1​​T)

T=\frac{v_1a}{{v_1}^2-{v_2}^2}T=​v​1​​​2​​−v​2​​​2​​​​v​1​​a​​

这样就很Simple地解完了，到达不了的情况就是v_1< v_2v​1​​<v​2​​（或者$a>0$且v_1=v_2v​1​​=v​2​​）。

 

#include<iostream>
#include<cstdio>
#include<cmath>
using namespace std;
int main()
{
    double a,x,y;
    while(cin>>a>>x>>y)
    {
        if(a==0)
        {
        printf("0\n");
        continue;
        }
        double ans=(a*x)/fabs(y*y-x*x);
        if(y>=x)
        printf("Infinity\n");
        
        else
        cout<<ans<<endl;
    }
}