Projectile Motion Calculator
Range (m)
40.8
Max height (m)
10.2
How it works
Projectile motion describes the two-dimensional trajectory of an object launched with an initial velocity at an angle, subject only to gravitational acceleration. The horizontal and vertical components are independent: horizontal velocity remains constant (no air resistance), while vertical velocity changes due to gravity (g = 9.81 m/s²).
**Key equations** Horizontal: x = v₀ × cos(θ) × t. Vertical: y = v₀ × sin(θ) × t - ½ × g × t². Maximum height: H = v₀² × sin²(θ) / (2g). Range: R = v₀² × sin(2θ) / g. Time of flight: T = 2 × v₀ × sin(θ) / g. Maximum range occurs at 45° launch angle.
**Effect of launch angle on range vs. height** At 45°, range is maximized. Angles symmetric about 45° give equal range — a 30° and 60° launch at the same speed reach the same horizontal distance, but the 60° launch reaches greater height and stays airborne longer. For artillery and sports, different angles suit different objectives.
**Air resistance in real projectiles** The idealized model neglects air drag. For dense, slow objects (shot put, baseball at moderate speed) the model is reasonably accurate. For light or high-speed projectiles (shuttlecock, bullet, golf ball), drag significantly shortens range and alters the optimal angle. The drag force is proportional to velocity squared, making aerodynamic effects dominant at high speed.
**Elevation difference** When launch and landing points are at different elevations, the range equation changes. Throwing downhill from a cliff increases range; throwing uphill reduces it. The calculator should account for initial height offset for realistic applications.
Frequently Asked Questions
- 45° maximizes range on level ground, giving R = v₀²/g. Angles symmetric about 45° give equal range: 30° and 60° both produce R = v₀² × sin(60°)/g = v₀² × (√3/2)/g. If the landing point is lower than the launch point (shooting off a cliff), the optimal angle is less than 45°. If landing is higher (shooting uphill), optimal angle is greater than 45°. In practice, air resistance reduces the optimal angle below 45° for most projectiles — artillery optimal angles are typically 30–42° depending on projectile drag.
- Air drag force = ½ × ρ × v² × C_d × A (density × velocity squared × drag coefficient × cross-sectional area). This acts opposite to velocity at every instant, reducing both horizontal and vertical velocity components. Effects: reduced range (often 20–60% less than vacuum model), lowered optimal angle (below 45°), asymmetric trajectory (steeper descent than ascent), and terminal velocity in the vertical direction. The vacuum model is reasonable for dense, slow objects (shot put, baseball at moderate speed) but seriously wrong for high-speed or light projectiles (bullets, golf balls, badminton shuttlecocks).
- Basketball free throw: optimal release angle ≈ 52° for a flat trajectory; higher angle increases probability of scoring by widening the effective target. Soccer goal kick: range equation estimates maximum distance; actual optimal angle is ~35° accounting for air drag. Golf: launch angle, backspin (generates Magnus lift), and ball speed all interact — driver optimal launch angle is ~12–14° with backspin of ~2500 RPM. Sports biomechanics uses high-speed cameras (1000+ fps) to measure actual launch conditions and compare to calculated optimal trajectories.
- Time to reach maximum height = v₀ × sin(θ) / g (half of total flight time, on level ground). Total time of flight T = 2 × v₀ × sin(θ) / g. Maximum height H = v₀² × sin²(θ) / (2g) = g × (T/2)² / 2. A projectile launched at 30 m/s at 60°: T = 2 × 30 × sin(60°) / 9.81 = 5.30 s; H = 30² × sin²(60°) / (2 × 9.81) = 34.4 m. The vertical component acts exactly like a ball thrown straight up — horizontal motion is completely independent.