Mass center of particles moving at constant velocity X30485
Preliminaries
In this exercise we will implement a program that prints real numbers on the output. Due to format and efficiency issues of cin and cout when they manage real numbers, it is convenient to do some things that we explain as follows.
At the beginning of function main you should place these instructions:
ios::sync_with_stdio(false); cin.tie(0); cout.setf(ios::fixed); cout.precision(5);
In order to print real numbers on the output, it is convenient to include and use this function:
void printDouble(double d)
{
if (abs(d) < 1e-7)
cout << 0.0;
else
cout << d;
}
Last but not least, do not use endl to print break lines. Use ’\n’ instead, that is:
// This line has been replaced with next one: // cout << endl; cout << '\n';
Exercise
Note: In this exercise we will speak about positions of particles in a 3-dimensional system, and about their velocities and masses. The reference and units system is not too relevant, as long as the chosen one is more or less standard and reasonable. Just in case someone feels the need, you can imagine that we speak about meters, seconds, meters per second and kilograms.
As input, we will have the position, velocity and mass of n particles (p1,v1,m1),…,(pn,vn,mn). In particular, the first particle is at position p1 at time 0, moves at constant velocity v1, and its mass is m1. Everything is supposed to be represented in a cartesian coordinate system with three dimensions.
Furthermore, we will have k values of time t1,…,tk as input.
We will have to determine the mass center of the n particles after t1 time units, after t1+t2 time units, after t1+t2+t3, …, after t1+⋯+tk time units.
It is compulsory to coherently use the following declarations of types, and implement and coherently use the following functions. Otherwise, the delivery will be invalidated. You can declare more data types and implement more functions if you feel like doing so. That’s actually advisable in this exercise.
struct Point {
double x, y, z;
};
struct Particle {
Point p,v;
double m;
};
// Pre:
// Post: returns the sum of p1 and p2.
Point sum(Point p1, Point p2)
{
//...
}
// Pre:
// Post: returns a times p.
Point mul(double a, Point p)
{
//...
}
Note: We strongly advise you to start doing a simple implementation in order to overcome the public tests, and to try to optimise it later on, in order to overcome the private tests as well.
Input
The input has several cases. Each one starts with two positive natural numbers n, k on a first line. Next, n lines come, each describing the position (three integers), velocity (three integers) and mass (a positive natural) of a particle. Finally, k lines come, each with a positive natural that represents an elapsed time.
Output
For each case, at first k lines must be printed, where i’th line contains the mass center (three reals rounded to 5 digits after the decimal point) of all the particles after the sum of the first i elapsed times. Secondly, n lines must be printed, with the positions (three reals rounded to 5 digits after the decimal point) of the particles after the sum of all the elapsed times. Each case is followed by a blank line.
Observation
Grading up to 10 points:
- Slow solution: 5 points.
- Fast solution: 10 points.
We understand as a fast solution one which is correct, with linear cost and which passes the public and private tests. We understand as slow solution one which is not fast, but it is correct and passes the public tests.
Input
2 2 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 2 2 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 2 2 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
Output
0.50000 0.00000 1.00000 0.50000 0.00000 2.00000 0.00000 0.00000 2.00000 1.00000 0.00000 2.00000 1.50000 0.00000 0.00000 2.50000 0.00000 0.00000 2.00000 0.00000 0.00000 3.00000 0.00000 0.00000 1.00000 0.00000 0.50000 1.50000 0.00000 1.00000 2.00000 0.00000 0.00000 1.00000 0.00000 2.00000
Input
4 2 1 -3 -4 -1 -5 1 5 -4 3 2 0 -2 2 2 4 5 -3 -5 5 3 3 -5 -1 1 -5 5 -2 1 5 3 3 4 2 4 2 3 -3 5 5 5 -1 -4 -5 5 -1 5 2 1 3 1 -3 -3 2 4 3 1 2 2 1 -1 -1 -2 1 4 5 5 -1 -4 -5 5 4 -1 5 3 5 5 1 4 2 -4 -1 4 4 4 -3 5 -2 2 1 1 4 3 4 -1 0 4 5 4 3 -2 3 2 0 2 -1 2 2 3 0 -4 2 1 2 2 1 5 5 5 -1 -3 -1 -2 5 -2 2 -3 5 3 5 0 -5 4 5 1 5 -2 -5 2 4 0 1 4 5 -1 4 4 -2 2 -3 3 2 -1 4 1 2 2 2 1 5 4 3 3 -5 2 -2 4 5 -4 0 3 -1 -5 1 2 -3 -3 4 1 2 4 3 -2 -2 3 3 3 4 1 -5 3 3 3 -1 1 4 2 4 5 2 5 4 -3 1 -2 -1 3 1 4 -4 -5 5 -5 -1 -5 5 5 2 2 -4 2 -3 1 -3 1 -3 -1 -4 0 5 3 5 -1 3 -2 1 5 4 2 3 4 5 4 -4 -4 -4 2 -5 3 1 -4 -3 -1 3 -1 5 2 3 4 -2 -3 1 -5 3 -3 -1 -5 -1 -2 3 2 -2 2 0 -3 4 -1 4 2 2 1 4 5 1 -1 0 -1 -3 -4 -5 4 -4 2 -1 1 4 1 5 -5 4 -4 0 -4 2 3 -3 -2 -3 -4 5 -4 4 1 -4 1 -5 -1 3 3 2 2 4 0 -3 -1 4 4 5 5 4 3 -3 -4 2 2 3 1 5 4 1 5 3 5 -2 3 -2 -4 -3 3 -3 0 1 -1 5 2 1 4 0 0 5 -3 -3 1 -4 2 2 4 1 -1 5 -4 -4 -4 -5 0 4 1 1 3 5 5 4 1 3 -2 -3 -2 5 5 3 4 2 3 2 2 5 -2 0 4 0 5 5 4 -5 4 5 4 5 -2 4 5 5 -4 -2 2 -3 2 4 5 1 2 4 3 -1 2 5 4 3 -3 4 -2 -4 -1 3 5 -5 4 2 -5 5 -1 3 2 1 -5 -3 0 -2 -5 2 3 5 3 5 4 5 -5 1 4 -2 -5 5 4 -2 0 -3 -2 -4 4 1 4 -5 4 1 -2 2 5 1 5 1 -2 4 -4 5 5 3 4 3 1 4 5 -1 -3 -4 0 2 -4 2 1 5 3 0 3 -1 2 -5 -4 -1 2 4 5 3 2 4 -1 0 3 -5 4 1 4 3 2 1 2 2 5 -4 -2 -2 4 4 1 2 -2 -1 0 -3 1 3 3 3 2 5 -5 4 4 3 3 3 5 5 1 5 -4 0 0 3 4 4 2 1 5 2 3 1 -5 -5 3 -1 4 3 3 -2 -5 3 -3 4 3 5 4 3 4 3 3 1 -1 -4 0 4 4 2 5 -5 -3 0 2 4 -2 -3 2 -4 2 -3 4 2 3 0 -2 -5 -3 3 1 3 2 3 1 -5 1 0 2 -4 5 2 -4 -5 2 1 -3 5 1 2 -4 -2 2 -1 -5 2 5
Output
-11.00000 -3.63636 5.09091 -17.81818 -6.09091 9.45455 -7.00000 -43.00000 4.00000 -4.00000 -13.00000 18.00000 -36.00000 45.00000 21.00000 -45.00000 39.00000 -15.00000 0.58333 2.75000 4.33333 -1.41667 3.75000 7.83333 -2.08333 4.08333 9.00000 -3.41667 4.75000 11.33333 32.00000 -26.00000 52.00000 -45.00000 49.00000 -14.00000 12.00000 -29.00000 -27.00000 8.00000 9.50000 2.50000 2.00000 11.00000 13.00000 14.00000 8.00000 -8.00000 1.64286 2.85714 4.42857 0.07143 3.14286 8.57143 -1.50000 3.42857 12.71429 -2.28571 3.57143 14.78571 -6.21429 4.28571 25.14286 -43.00000 -7.00000 46.00000 -18.00000 19.00000 16.00000 -7.00000 3.00000 48.00000 37.00000 25.00000 -2.00000 32.00000 2.00000 -42.00000 2.11111 1.33333 1.66667 6.55556 0.66667 1.22222 11.00000 0.00000 0.77778 15.44444 -0.66667 0.33333 17.66667 -1.00000 0.11111 -17.00000 37.00000 -17.00000 37.00000 5.00000 -37.00000 -11.00000 -39.00000 21.00000 40.00000 -7.00000 36.00000 22.00000 18.00000 -11.00000 1.86667 -0.93333 6.13333 8.80000 -4.93333 17.33333 17.46667 -9.93333 31.33333 20.93333 -11.93333 36.93333 29.00000 -23.00000 47.00000 -17.00000 -65.00000 16.00000 10.00000 23.00000 56.00000 37.00000 37.00000 55.00000 34.00000 -10.00000 16.00000 -5.95000 -3.65000 -3.85000 -8.25000 -5.75000 -5.75000 -11.70000 -8.90000 -8.60000 -16.30000 -13.10000 -12.40000 -16.00000 40.00000 11.00000 -69.00000 -18.00000 -60.00000 -47.00000 28.00000 -37.00000 -16.00000 -51.00000 -3.00000 42.00000 -21.00000 12.00000 -3.91667 2.00000 -1.83333 -6.41667 3.33333 -1.83333 -7.66667 4.00000 -1.83333 -12.66667 6.66667 -1.83333 14.00000 -49.00000 23.00000 23.00000 -12.00000 44.00000 -24.00000 13.00000 -47.00000 -12.00000 -19.00000 22.00000 -29.00000 38.00000 -9.00000 -6.52632 1.05263 -3.26316 -7.00000 -8.00000 -11.00000 -2.00000 10.00000 1.00000 -5.00000 -4.00000 0.00000 -11.00000 8.00000 -11.00000 -9.00000 -6.00000 7.00000 2.50000 2.50000 2.12500 7.50000 18.75000 21.50000 11.50000 31.75000 37.00000 12.50000 35.00000 40.87500 44.00000 41.00000 54.00000 -40.00000 25.00000 19.00000 0.45455 -0.45455 0.45455 4.00000 -1.81818 -2.54545 9.90909 -4.09091 -7.54545 -13.00000 -38.00000 -24.00000 -12.00000 45.00000 19.00000 49.00000 -27.00000 -27.00000 32.00000 11.00000 -7.00000 -49.00000 -4.00000 32.00000 2.50000 11.85000 9.60000 5.50000 22.85000 19.85000 6.10000 25.05000 21.90000 7.30000 29.45000 26.00000 -35.00000 -21.00000 58.00000 39.00000 28.00000 26.00000 -2.00000 60.00000 64.00000 43.00000 64.00000 -19.00000 -19.00000 29.00000 -40.00000 6.66667 6.50000 -8.25000 11.91667 11.50000 -14.25000 20.66667 19.83333 -24.25000 51.00000 41.00000 -34.00000 37.00000 61.00000 -66.00000 -11.00000 34.00000 31.00000 -31.00000 -68.00000 26.00000 -4.80000 -4.40000 7.20000 -7.80000 -7.20000 10.00000 -11.80000 -10.93333 13.73333 -14.80000 -13.73333 16.53333 -15.80000 -14.66667 17.46667 -37.00000 -79.00000 84.00000 -34.00000 -64.00000 61.00000 20.00000 -37.00000 36.00000 -31.00000 69.00000 -63.00000 -0.09091 3.81818 -2.18182 2.09091 16.18182 -7.63636 3.72727 25.45455 -11.72727 4.81818 31.63636 -14.45455 5.36364 34.72727 -15.81818 -1.00000 19.00000 -48.00000 1.00000 38.00000 -8.00000 17.00000 40.00000 54.00000 2.00000 37.00000 -56.00000 1.25000 -6.25000 4.00000 -0.25000 -10.00000 9.25000 -7.00000 20.00000 22.00000 2.00000 -20.00000 5.00000 0.25000 10.37500 11.87500 1.75000 17.87500 19.37500 2.50000 21.62500 23.12500 2.87500 23.50000 25.00000 4.75000 32.87500 34.37500 43.00000 52.00000 52.00000 -59.00000 1.00000 5.00000 17.00000 -13.50000 15.00000 29.00000 -21.50000 31.00000 38.00000 -27.50000 43.00000 37.00000 -17.00000 43.00000 39.00000 -38.00000 43.00000 -2.13333 0.93333 -0.86667 -12.13333 -0.46667 -0.26667 -18.80000 -1.40000 0.13333 -21.00000 1.00000 23.00000 -16.00000 5.00000 7.00000 -26.00000 9.00000 -16.00000 -10.00000 -27.00000 -18.00000 7.00000 -15.20000 4.60000 5.00000 -19.00000 25.00000 1.00000 -20.00000 27.00000 12.00000 -9.00000 -27.00000