Projectile Range
Calculate horizontal range for projectiles. Find optimal launch angle and compare range at different angles.
Did our AI summary help? Let us know.
Why: Understanding projectile range helps you make better, data-driven decisions.
How: Enter Initial Velocity (m/s), Launch Angle (°), Initial Height (m) to calculate results.
Run the calculator when you are ready.
Projectile Range Calculator
R = v₀²sin(2θ)/g • Optimal 45° • Trajectory • Range vs Angle
Sample Scenarios — Click to Load
Input Parameters
📈 Visualizations
Projectile Trajectory
Range vs Launch Angle
Range Comparison
📝 Step-by-Step Solution
Initial velocity: v₀ = 20.00 m/s
Launch angle: θ = 45.0°
Initial height: h₀ = 0.00 m
Gravity: g = 9.81 m/s²
Vertical: v₀ᵧ = v₀sin(θ)
→ v₀ᵧ = 14.14 m/s
Horizontal: v₀ₓ = v₀cos(θ)
→ v₀ₓ = 14.14 m/s
Time of flight: t = (v₀ᵧ + √(v₀ᵧ² + 2gh₀)) / g
→ t = 2.883 s
Range: R = v₀ₓ × t
R = 14.1421 × 2.8832
→ R = 40.77 m
Maximum height
→ H_max = 10.19 m
Optimal launch angle for max range
→ θ_optimal = 45.0°
Maximum possible range at optimal angle
→ R_max = 40.77 m
Impact velocity
→ v_impact = 20.00 m/s
Impact angle
→ θ_impact = 45.0° below horizontal
For educational and informational purposes only. Verify with a qualified professional.
📋 Key Takeaways
- • R = v₀²sin(2θ)/g — range on level ground; 45° maximizes range
- • Complementary angles (θ and 90°−θ) give same range
- • Range ∝ v₀² — doubling speed quadruples range
- • Elevated launch: optimal angle < 45°; R = v₀ₓ(v₀ᵧ+√(v₀ᵧ²+2gh₀))/g
💡 Did You Know?
📖 How Projectile Range Works
Range is horizontal distance traveled. x = v₀cos(θ)t; y = v₀sin(θ)t − ½gt². At landing y=0 → t = 2v₀sin(θ)/g → R = v₀²sin(2θ)/g.
Step 1: Velocity components
v₀ₓ = v₀cos(θ), v₀ᵧ = v₀sin(θ)
Step 2: Flight time
t = (v₀ᵧ + √(v₀ᵧ² + 2gh₀))/g
Step 3: Range
R = v₀ₓ × t
🎯 Expert Tips
💡 45° Rule
45° optimal only for level ground. Elevated launch → θ_opt < 45°.
💡 Air Resistance
Real projectiles: optimal 30–40°. Golf ~12° due to lift.
💡 v₀² Effect
Double velocity = 4× range. Speed matters most.
💡 Complementary Angles
30° and 60° give same range; different trajectories.
⚖️ Range Comparison
| v₀ | 15° | 30° | 45° | 60° |
|---|---|---|---|---|
| 20 m/s | 20.4 | 35.3 | 40.8 | 35.3 |
| 50 m/s | 127.6 | 220.8 | 255.1 | 220.8 |
❓ FAQ
Why 45° optimal?
R = v₀²sin(2θ)/g. sin(2θ) max at 2θ=90° → θ=45°. Level ground only.
Air resistance effect?
Reduces range; shifts optimal to 30–40°. Golf ~12° due to lift.
Elevated launch?
Optimal angle < 45°. Extra falling time increases range.
Doubling velocity?
R ∝ v₀². Double v₀ → 4× range.
Complementary angles?
θ and 90°−θ give same sin(2θ), hence same range.
Different planets?
R ∝ 1/g. Moon ~6× Earth; Mars ~2.6× Earth.
📊 By the Numbers
📚 Sources
⚠️ Disclaimer: Ideal projectile motion (no air resistance, uniform g). Real projectiles experience drag, wind, spin. Use ballistics calculators for firearms/artillery.
Related Calculators
Time of Flight Calculator
Calculate projectile flight duration. Find time to apex, total flight time, and trajectory details.
PhysicsBallistic Coefficient Calculator
Calculate ballistic coefficient for projectiles. Analyze G1, G7 drag models, sectional density, and trajectory performance.
PhysicsHorizontal Projectile Motion Calculator
Calculate trajectory, flight time, and impact velocity for objects launched horizontally.
PhysicsMaximum Height Calculator
Calculate maximum height of projectiles. Analyze trajectory, time to apex, and flight path.
PhysicsMuzzle Velocity Calculator
Calculate muzzle velocity for firearms and projectiles. Analyze kinetic energy, momentum, and ballistics.
PhysicsMomentum Calculator
Calculate momentum (p = mv), impulse, and analyze collisions. Understand conservation of momentum in physics.
Physics