Projectile Range Calculator
Calculate maximum horizontal distance of projectile motion
Calculate Projectile Range
Speed at which projectile is launched
Angle above horizontal (45° = optimal)
Height above ground at launch
Range Calculation Results
Velocity Components
Range Analysis
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
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
Basketball Shot
Long-range basketball shot
Cannonball (Historical)
Medieval artillery
Javelin Throw
Olympic javelin throw
Range Optimization
Optimal Angle
45° gives maximum range on level ground
Velocity Impact
Range increases with velocity squared
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