// calculates the velocity required and earth time passed to travel to a distant
// place at relativistic speeds with some moving reference frame travel time.
// e.g. if it takes you 3 years travel time to travel 50lyr, what is your velocity
// and how much earth time has passed before you get there.


#include <iostream>
#include <math.h>
#include <fstream>
#include <iomanip>

using namespace std;

double travelTime, Distance, velocity, earthTime;
double tol = 0.0000001;

double func(double v)
{
	return (Distance / v) - (travelTime / sqrt(1-pow(v,2.0))); //setting c = 1
}

double deriv(double v)
{
	return (-Distance / pow(v,2.0)) + (0.5*travelTime / pow(1-pow(v,2.0),1.5))*(-2.0*v);
}

int main()
{
	cout << "Travel Time (years) = ";
	cin >> travelTime;
	cout << "Distance (light-years) = ";
	cin >> Distance;
	
	velocity = 0.9; //initial guess
	int i = 1000000.0; //maximum iterations after which assume v = c
	do //NR Method
	{
		velocity -= func(velocity) / deriv(velocity); 
		if (velocity >= 1.0){velocity = 1.0-tol;}
		i -= 1.0;
	} while ((fabs(func(velocity)) >= tol) and (i > 0));
	
	if (i==0)
	{
		earthTime = Distance;
	}
	else  if (velocity < 0)
	{
		velocity = 0;
		earthTime = travelTime;
	}
	else
	{
		earthTime = (travelTime / sqrt(1-pow(velocity,2.0)));
	}

	cout << "Velocity = " << velocity << " c" << endl;
	cout << "Earth Time = " << earthTime << " years" << endl;
		
	return 0;
}