From archers in the ancient world to space rocket launches today, understanding projectile motion is one of the greatest triumphs of physics.

When an object is launched with initial speed $v_0$ at an angle $\theta$ above the horizontal:

• Horizontal Motion ($x$): In the absence of air friction, no horizontal forces act on the projectile. Its horizontal velocity remains completely constant: $$v_x = v_0 \cos\theta \implies x(t) = x_0 + v_x \cdot t$$
• Vertical Motion ($y$): Gravity constantly accelerates the projectile downward with acceleration $g$: $$v_y(t) = v_0 \sin\theta - g \cdot t \implies y(t) = y_0 + (v_0 \sin\theta) \cdot t - \frac{1}{2}g t^2$$

The Parabolic Path

Eliminating time $t$ from the equations reveals that the trajectory is a downward-opening parabola: $$y(x) = x \tan\theta - \frac{g}{2 v_0^2 \cos^2\theta} x^2$$

Maximum Range and Complementary Angles

The total horizontal landing distance (range $R$) over flat ground is: $$R = \frac{v_0^2 \sin(2\theta)}{g}$$

• Because $\sin(2\theta)$ reaches its absolute peak of $1.0$ when $2\theta = 90^\circ$, the angle that produces the maximum possible range is $\theta = 45^\circ$!
• Two launch angles that sum to $90^\circ$ (called complementary angles, like $30^\circ$ and $60^\circ$, or $25^\circ$ and $65^\circ$) land at the exact same distance because $\sin(2\theta) = \sin(180^\circ - 2\theta)$!

To simulate projectiles in code:

• add_floor() creates the ground surface.
• add_box(x, y, w, h, is_static, angle, color) places a rectangular obstacle or wall.
• launch_ball(x, y, vx, vy, radius, restitution, color) launches a projectile with initial horizontal ($v_x$) and vertical ($v_y$) velocities.
• physics_world(gx, gy) configures custom world gravity (default is Earth gravity (0, 1)).

Move the mouse over a dotted box for more information.

• Coordinate Signs: Screen pixels measure $y = 0$ at the top of the canvas and $y = 400$ at the bottom. Therefore, launching upward requires a negative vertical velocity ($v_y = -v_0 \sin\theta$).
• Key Trajectory Formulas:
  • Peak height: $H = \frac{v_0^2 \sin^2\theta}{2g}$
  • Total time of flight: $T = \frac{2 v_0 \sin\theta}{g}$
  • Horizontal range: $R = \frac{v_0^2 \sin(2\theta)}{g}$
• Interactive Fun: Modify the launch angle and speed in the code editor to lob balls over obstacles or into targets!