Projectile Range

Because sin(2θ) = sin(180° − 2θ). Setting 2θ₁ = 180° − 2θ₂ gives θ₂ = 90° − θ₁. So any two complementary angles — like 30° and 60°, or 15° and 75° — produce identical ranges. This symmetry is a mathematical consequence of the shape of the sine curve, but it also has a physical reading: a 30° launch throws the projectile nearly flat, so it moves fast but lands quickly. A 60° launch throws it steeply, so it moves slowly but stays aloft much longer. The two effects cancel exactly in the vacuum approximation. In real air the symmetry breaks, because drag depends on speed and the flatter trajectory loses more energy.

In the vacuum approximation, no. Mass cancels out of the equations entirely: the horizontal and vertical motions are independent of mass, and the trajectory depends only on the launch speed and angle. A 1 g pebble and a 1 kg cannonball launched at the same speed and angle travel exactly the same distance. In the real world with air resistance, mass matters a great deal — a heavier projectile has more momentum for the same speed, so drag decelerates it less over a given distance, and it retains more of its speed throughout the flight. This is why lead shot outranges equal-size plastic pellets and why the same cannon firing a heavier shell achieves a longer range.

Air resistance reduces the range, flattens the optimal launch angle, and breaks the complementary-angle symmetry. The drag force is proportional to v² at typical projectile speeds, so a fast projectile loses energy at a rate that grows rapidly with speed. The result is that the projectile's actual path lies entirely inside the vacuum parabola, landing closer and at a shallower impact angle. The magnitude of the effect is captured by the ballistic parameter β = ρ C_d A v² / (mg), which compares the drag force to gravity. For β ≪ 1 (heavy, slow, dense projectiles like cannonballs at moderate speed) the vacuum formula is accurate to a few percent. For β ≳ 0.1 (light, fast, draggy projectiles like golf balls and shuttlecocks) the vacuum range can be 40% too high.

Because drag pushes it downward. A flatter trajectory spends less time in the air and passes through the fast-moving early portion of the flight more quickly, so it loses less total energy to drag. For a baseball thrown at typical outfielder speeds, the optimum is around 35°. For a golf ball with backspin — which actually generates lift — the optimal launch angle can be as low as 12°. For a heavy artillery shell the optimum is around 40°. But the launch conditions also matter: an athlete cannot always achieve the ideal angle given the constraints of their body and the equipment, and the optimal angle for a kicker or a thrower also depends on the ability to impart speed at different angles.

The formula R = v² sin(2θ)/g assumes the projectile launches from and lands at the same height. If it lands on higher ground, the range is shorter than the formula predicts. If it lands in a valley, the range is longer. The general formula for a projectile launched from height h above the landing surface is R = (v cos θ/g) · (v sin θ + √(v² sin²θ + 2gh)). This reduces to the simple form when h = 0. For a cliff launch, the projectile stays aloft longer, and the maximum range occurs at an angle slightly less than 45° — the effect is small for shallow h but significant for launches high above the landing surface.

For a projectile launched and landing at the same level, the maximum height is H = v² sin²θ/(2g) and the range is R = v² sin(2θ)/g. Their ratio is H/R = sin²θ / (2 sin 2θ) = sin²θ / (4 sin θ cos θ) = tan θ / 4. So H = R · tan θ / 4. At 45° this gives H = R/4, meaning the projectile peaks at one quarter of its range — the apex is fairly close to the landing point. At 30° the apex is only R · 0.144, very flat. At 60° the apex is R · 0.433, noticeably taller. At the extremes — θ → 0° or θ → 90° — the range goes to zero and the ratio becomes undefined or infinite.

Only as a rough first estimate. The formula assumes constant gravity, no air resistance, no lift, and a ballistic trajectory with no thrust after launch. A rocket is under thrust for part of its flight, so its trajectory is not a parabola and the range must be computed by integrating the equations of motion with a time-varying thrust. A spinning ball experiences the Magnus force, which can produce lift (backspin) or downward force (topspin), and the trajectory is no longer a parabola. A golf ball with backspin can travel 50–60% farther than the vacuum formula predicts. For any of these cases the vacuum formula is a useful starting point but the real trajectory must be computed numerically.