Question 1
Neglecting air resistance, a projectile fired from a gun follows a parabola in the vertical plane through the barrel axis. Take the gun's ground position as $O$, the horizontal line as $Ox$ and the vertical line as $Oy$. Its trajectory is
$$y=-\frac{g}{2v_0^2\cos^2\alpha}x^2+x\tan\alpha,$$
where $\alpha$ is the barrel's angle with the ground, $g$ is gravitational acceleration, and $v_0$ is the initial speed.
(a) For which angle does the projectile travel farthest?
(b) For which angles does it hit a point with coordinates $X\ge0$, $Y\ge0$? What region of space consists of reachable points?
$$y=-\frac{g}{2v_0^2\cos^2\alpha}x^2+x\tan\alpha,$$
where $\alpha$ is the barrel's angle with the ground, $g$ is gravitational acceleration, and $v_0$ is the initial speed.
(a) For which angle does the projectile travel farthest?
(b) For which angles does it hit a point with coordinates $X\ge0$, $Y\ge0$? What region of space consists of reachable points?