Projectile Range Calculator

Calculate maximum horizontal distance of projectile motion

Calculate Projectile Range

m/s

Speed at which projectile is launched

°

Angle above horizontal (45° = optimal)

m

Height above ground at launch

Range Calculation Results

40.77
Range (m)
2.88
Flight Time (s)
10.19
Max Height (m)
100.0%
Range Efficiency

Velocity Components

Horizontal (vₓ):14.14 m/s
Vertical (vᵧ):14.14 m/s

Range Analysis

Current Range:40.77 m
Max Possible Range:40.77 m
Efficiency:100.0%

Physics Formulas Used

Ground Level Launch: R = v₀²sin(2α)/g

Optimal Angle: 45° gives maximum range

Time of Flight: t = 2v₀sin(α)/g

Range Analysis

✅ Optimal angle for maximum range (45°)
🎯 Excellent range efficiency (>90%)

Example: Soccer Ball Kick

Problem Setup

Scenario: Professional soccer player kicks ball from ground level

Given: v₀ = 25 m/s, α = 25°, h = 0 m

Question: How far will the ball travel?

Solution Steps

1. Use ground-level formula: R = v₀²sin(2α)/g

2. Calculate: R = (25)² × sin(2 × 25°) / 9.81

3. R = 625 × sin(50°) / 9.81 = 625 × 0.766 / 9.81

4. R = 478.75 / 9.81 = 48.8 meters

Result: The ball travels 48.8 meters horizontally

Real-World Examples

Soccer Ball

Professional soccer kick

Velocity: 25 m/s
Angle: 25°
Height: 0 m

Basketball Shot

Long-range basketball shot

Velocity: 8 m/s
Angle: 45°
Height: 2 m

Cannonball (Historical)

Medieval artillery

Velocity: 50 m/s
Angle: 45°
Height: 5 m

Javelin Throw

Olympic javelin throw

Velocity: 30 m/s
Angle: 35°
Height: 2 m

Range Optimization

45°

Optimal Angle

45° gives maximum range on level ground

Velocity Impact

Range increases with velocity squared

h

Height Advantage

Launch height increases range

Complementary Angles

Angles 30° and 60° give same range

Essential Range Formulas

Ground Level Launch

R = v₀²sin(2α)/g

Maximum at α = 45°

Elevated Launch

R = vₓ × t_flight

Where t = (vᵧ + √(vᵧ² + 2gh))/g

Velocity Components

vₓ = v₀cos(α)

vᵧ = v₀sin(α)

Maximum Range

R_max = v₀²/g

Theoretical maximum at 45°

Understanding Projectile Range

What is Projectile Range?

Projectile range is the horizontal distance traveled by an object launched into the air before it returns to the same height from which it was launched. It's one of the most important parameters in ballistics and sports physics.

Factors Affecting Range

Range depends on initial velocity (squared relationship), launch angle, and initial height. Air resistance is typically ignored in basic calculations but significantly affects real-world projectiles.

Why 45° is Optimal

For ground-level launches, 45° provides the optimal balance between horizontal and vertical velocity components, maximizing the sin(2α) term in the range formula.

Practical Applications

Range calculations are crucial in sports (basketball, soccer, javelin), military ballistics, engineering design, and space missions for trajectory optimization.

Sports Applications

  • • Soccer ball trajectory analysis
  • • Basketball shot optimization
  • • Javelin throw mechanics
  • • Golf ball flight paths

Engineering Uses

  • • Artillery range calculations
  • • Water fountain design
  • • Spacecraft trajectory planning
  • • Safety barrier placement

Physics Education

  • • Kinematic motion analysis
  • • Trigonometric applications
  • • Optimization problems
  • • Vector decomposition