Background

Description

After returning to class and seeing his 28/110 in physics, little F felt utterly defeated. He decides to start learning mechanics from the very basics.

As shown in the figure, a small ball that can be treated as a point mass is thrown from point $A(x_0, y_0)$ along the negative direction of the $x$-axis with some speed, ignoring all resistance except gravity, and finally hits point $B(0, 0)$ exactly with speed $v$.

Given the magnitude and direction of $v$, your task is to find $(x_0, y_0)$.

The unit of the given speed is $m \cdot s ^ {-1}$, and the gravitational acceleration is $g = 10 \ (m \cdot s ^ {-2})$. Please output the answer in meters.

If you have not learned the related content, it is okay; you can understand what is required from the samples and the hints.

Output Format

Output one line with two real numbers $x_0, y_0$ (each with at most $15$ decimal places), which are your answers.

Your answer is considered correct if the absolute or relative error is less than $10 ^ {-3}$.

14.142136 0.785398

10 5

Hint

Sample Explanation

As shown.

$14.142136 \approx 10 \sqrt 2, 0.785398 \approx \frac \pi 4 = 45 ^ \circ.$

If the ball is thrown from $(10, 5)$ with velocity $(-10, 0)$, then at $t = 1 s$ it hits $(0, 0)$ with velocity $(-10, -10)$.

Hint

If you have not studied the related content, the following may help:

zcy teaches you physics

First, since all units are standard, all results can be computed directly with numbers; treating the object as a point mass means it has no volume.

We can decompose the velocity of the ball as shown:

The horizontal component of the velocity $v_x$ is the initial throw speed and remains $v \sin \theta$ during the motion.

The vertical component of the velocity $v_y$ is accelerated by gravity, changing from $0$ to $v \cos \theta$.

Starting timing from the moment of release, when the time is $t$, let the magnitudes of the horizontal and vertical velocities at this moment be $v_{xt}, v_{yt}$, and the magnitudes of the horizontal and vertical displacements be $x_{xt}, x_{yt}$, respectively. Then:

$v_{xt} = v \sin \theta$

$v_{yt} = gt$

$x_{xt} = v_{xt}t$

$x_{yt} = \frac 1 2 g t ^ 2 = \frac 1 2{v_{yt}t}$

When $t$ is exactly the landing time, $x_{xt}, x_{yt}$ are the answers.

About radians:

$\pi = 180 ^{\circ}$

That is: $\frac \pi 2 = 90 ^{\circ}, \frac \pi 3 = 60 ^{\circ}, \ \cdots$

About trigonometric functions:

If you use C/C++, you can use sin() and cos() from math.h / cmath.

If you use Pascal, you can use sin() and cos() from the math library (add uses math; before begin).

If you use Python, you can use math.sin() and math.cos() from the math library.

If you use other languages, please refer to the corresponding documentation.

Translated by ChatGPT 5