Projectile Motion

The setup

An object is launched at speed v₀ at angle θ above horizontal. Only gravity acts (ignoring air resistance for now).

Decompose initial velocity into components:

• Horizontal: v_x = v₀·cos θ.
• Vertical: v_y₀ = v₀·sin θ.

These two motions are independent. Solve each separately.

Equations of motion

Quantity	Horizontal	Vertical
Acceleration	a_x = 0	a_y = -g
Velocity	v_x = v₀·cos θ (constant)	v_y = v₀·sin θ − g·t
Position	x = v₀·cos θ · t	y = v₀·sin θ · t − ½g·t²

Trajectory — a parabola

Eliminate t from the equations: from horizontal, t = x / (v₀·cos θ). Substitute into vertical:

y = x·tan θ − g·x² / (2·v₀²·cos²θ)

This is a quadratic in x — a parabola opening downward. The trajectory is always parabolic for projectile motion under uniform gravity (no drag).

Key results

Time of flight (returning to same height):

T = 2·v₀·sin θ / g

Maximum height:

H = (v₀·sin θ)² / (2g)

Range (horizontal distance to return to launch height):

R = v₀²·sin(2θ) / g

Maximum range at θ = 45°: R_max = v₀²/g.

v₀	θ = 30°	θ = 45° (max)	θ = 60°
20 m/s	R = 35.3 m	R = 40.8 m	R = 35.3 m
30 m/s	R = 79.5 m	R = 91.7 m	R = 79.5 m
50 m/s	R = 220.8 m	R = 254.8 m	R = 220.8 m

Notice — 30° and 60° give the same range (since sin(60°) = sin(120°)). The angles are complements summing to 90°.

Non-level launches

If launching from a height h above the landing point:

R = (v₀·cos θ / g) · [v₀·sin θ + √((v₀·sin θ)² + 2g·h)]

Optimum launch angle is less than 45° in this case.

With air resistance — the real world

Drag force ~ ½ρ·C_d·A·v² (quadratic at high speeds). This makes the trajectory non-parabolic — flatter on the way up, steeper on the way down. Range is reduced; max height is reduced; optimum angle is below 45° (~30-40°).

For real projectiles:

• Cannonballs (large, slow) — close to parabolic.
• Bullets (fast, small) — significant drag; trajectory falls faster than expected.
• Baseballs (size, drag, spin matters) — Magnus effect from spin curves trajectory.
• Tennis balls — high drag (fuzzy surface); slow easily.
• Golf balls — dimples reduce drag at high Reynolds number; backspin gives lift.

JavaScript — projectile simulation

// Analytical projectile (no drag)
function projectile(v0, angle_deg, g = 9.81) {
  const angle = angle_deg * Math.PI / 180;
  const vx = v0 * Math.cos(angle);
  const vy0 = v0 * Math.sin(angle);
  const T = 2 * vy0 / g;
  const R = v0 * v0 * Math.sin(2 * angle) / g;
  const H = vy0 * vy0 / (2 * g);
  return { time_of_flight: T, range: R, max_height: H };
}

console.log(projectile(30, 45));  // 30 m/s at 45°: R = 91.7 m
console.log(projectile(30, 30));  // 30 m/s at 30°: R = 79.5 m
console.log(projectile(30, 60));  // 30 m/s at 60°: R = 79.5 m (same as 30°!)

// Numerical simulation with optional drag
function simulate(v0, angle_deg, mass = 1, drag_coef = 0, dt = 0.01) {
  const angle = angle_deg * Math.PI / 180;
  let x = 0, y = 0;
  let vx = v0 * Math.cos(angle);
  let vy = v0 * Math.sin(angle);
  const trajectory = [{ x, y }];
  while (y >= 0 || x === 0) {
    // Forces: gravity + quadratic drag
    const v = Math.sqrt(vx*vx + vy*vy);
    const dragX = -drag_coef * v * vx;
    const dragY = -drag_coef * v * vy;
    const ax = dragX / mass;
    const ay = -9.81 + dragY / mass;
    vx += ax * dt;
    vy += ay * dt;
    x += vx * dt;
    y += vy * dt;
    if (y >= 0) trajectory.push({ x, y });
    if (trajectory.length > 10000) break;
  }
  return trajectory;
}

// No drag: should match analytical
const traj_nodrag = simulate(30, 45, 1, 0);
console.log(`No drag range: ${traj_nodrag[traj_nodrag.length-1].x.toFixed(1)} m`); // ~91.7

// With drag
const traj_drag = simulate(30, 45, 1, 0.005);
console.log(`With drag range: ${traj_drag[traj_drag.length-1].x.toFixed(1)} m`); // less

// Find optimum angle (no drag) — should be 45°
function findOptimumAngle(v0) {
  let best = 0, bestRange = 0;
  for (let a = 1; a < 90; a += 0.5) {
    const { range } = projectile(v0, a);
    if (range > bestRange) {
      bestRange = range;
      best = a;
    }
  }
  return { angle: best, range: bestRange };
}

console.log(findOptimumAngle(30));  // {angle: 45, range: 91.74}

Where projectile motion shows up

• Sports. Football kicks, basketball shots, baseball pitches/home runs, golf drives, javelin, shot put, tennis serves.
• Military and ballistics. Artillery shells, mortar rounds, missile trajectories. Real-world ballistic computation includes drag, wind, Earth's rotation.
• Cinema and animation. Realistic falling/throwing requires projectile equations. Most physics engines integrate them per frame.
• Spacecraft. Sub-orbital trajectories follow projectile motion. Reentry trajectories use modified projectile + drag analyses.
• Forensics. Bullet trajectory reconstruction, blood spatter analysis.
• Civil engineering. Water fountains, irrigation jets, fireworks design.
• Education. Classic intro physics problem demonstrating independent components, vector decomposition, and energy conservation.

Common mistakes

• Forgetting horizontal and vertical are independent. They don't affect each other. Don't try to apply gravity horizontally.
• Ignoring air resistance for fast/large objects. Bullets, baseballs, soccer balls — all need drag corrections. Pure parabolic is an idealization.
• Confusing "max range" angle with "max height" angle. Max range at 45° (no drag); max height at 90° (vertical, no horizontal). Different optimums.
• Wrong launch height handling. If launching from a cliff or a hill, range formula differs. Be careful — symmetric formulas R = v²·sin(2θ)/g assume same launch and landing heights.
• Mixing up sin and cos. Horizontal: v·cos θ. Vertical: v·sin θ. Easy mistake at 45° where they're equal, but matters at other angles.
• Forgetting that "projectile" means no thrust. Once a rocket cuts engines, it's a projectile. While engines are firing, additional forces apply.