Projectile Range Calculator
Calculate the range and trajectory parameters for projectile motion
Projectile range is the horizontal distance traveled by a projectile from its initial position to where it lands. In ideal conditions (neglecting air resistance), this distance depends primarily on:
- Initial velocity of the projectile
- Angle of launch
- Initial height (if any)
- Acceleration due to gravity
Understanding projectile range is crucial in many applications, from sports to military ballistics.
Initial Velocity
The initial velocity (v₀) directly affects the range. A higher initial velocity results in a greater range, with the relationship being quadratic (R ∝ v₀²).
Launch Angle
The launch angle (θ) determines how the initial velocity is split between horizontal and vertical components. For a flat surface:
- 45° provides the maximum range
- Complementary angles (e.g., 30° and 60°) give equal ranges
- 0° or 90° result in zero range
Range Equation
R = (v₀² × sin(2θ)) / g
Where: R = range, v₀ = initial velocity, θ = launch angle, g = acceleration due to gravity
Maximum Height
h_max = (v₀² × sin²(θ)) / (2g)
Time of Flight
t = (2 × v₀ × sin(θ)) / g
Maximum Range
R_max = v₀² / g (at θ = 45°)
Example 1: Maximum Range
A ball is launched at 20 m/s at a 45° angle. Calculate the range:
- Using R = (v₀² × sin(2θ)) / g
- R = (20² × sin(90°)) / 9.81
- R = 40.8 meters
Example 2: Different Angles
Compare ranges for the same ball (20 m/s) at different angles:
- At 30°: 35.3 meters
- At 45°: 40.8 meters (maximum)
- At 60°: 35.3 meters
Why is 45° the optimal angle for maximum range?
45° provides the perfect balance between horizontal and vertical velocity components. This maximizes the time of flight while maintaining sufficient horizontal motion.
How does air resistance affect range?
Air resistance reduces the actual range compared to calculated values. The effect increases with velocity and is more significant for lighter or less aerodynamic objects.
What factors can increase range?
Range can be increased by: increasing initial velocity, optimizing launch angle (usually 45°), launching from a height, or reducing air resistance through better aerodynamics.