🎯 Projectile Range Calculator
Calculate projectile range
How to Use This Calculator
Enter Initial Velocity
Input the initial speed at which the projectile is launched, measured in meters per second (m/s). This is the magnitude of the velocity vector.
Enter Launch Angle
Input the angle at which the projectile is launched above the horizontal, measured in degrees. For maximum range (without air resistance), use 45°.
Set Gravity (Optional)
The default gravity is 9.81 m/s² (Earth). You can change this for calculations on other planets or different gravitational fields.
Click Calculate
Press the "Calculate" button to compute the horizontal range and see the maximum possible range (at 45°).
Formula
R = (v₀² sin(2θ)) / g
Maximum Range (at θ = 45°): R_max = v₀² / g
Where:
R = Horizontal range (m)
v₀ = Initial velocity (m/s)
θ = Launch angle (degrees)
g = Gravitational acceleration (m/s²)
sin(2θ) = sin(2 × θ in radians)
Note: This formula assumes launch from ground level (h₀ = 0)
Example 1: Maximum range at 45°
Given: v₀ = 30 m/s, θ = 45°, g = 9.81 m/s²
Step 1: Calculate sin(2θ)
sin(2 × 45°) = sin(90°) = 1
Step 2: Apply range formula
R = (30² × 1) / 9.81 = 900 / 9.81 = 91.74 m
Or using maximum range formula: R_max = 30² / 9.81 = 91.74 m
Example 2: Range at 30°
Given: v₀ = 30 m/s, θ = 30°, g = 9.81 m/s²
sin(2 × 30°) = sin(60°) = 0.866
R = (30² × 0.866) / 9.81 = 779.4 / 9.81 = 79.45 m
Note: Range is less than maximum because angle is not 45°
Example 3: Range at 60°
Given: v₀ = 30 m/s, θ = 60°, g = 9.81 m/s²
sin(2 × 60°) = sin(120°) = 0.866
R = (30² × 0.866) / 9.81 = 79.45 m
Note: 30° and 60° give the same range! This is because sin(2×30°) = sin(2×60°)
About Projectile Range Calculator
The Projectile Range Calculator is a specialized physics tool for calculating the horizontal distance traveled by a projectile launched at an angle. The range is the horizontal distance from the launch point to where the projectile lands. This calculator uses the fundamental projectile motion formula R = (v₀² sin(2θ)) / g, which shows that range depends on the initial velocity squared, the sine of twice the launch angle, and gravitational acceleration. The calculator also displays the maximum possible range, which occurs at a 45° launch angle (for projectiles launched from ground level).
When to Use This Calculator
- Physics Problems: Calculate horizontal range for projectile motion problems
- Sports Analysis: Determine how far a ball will travel when kicked or thrown at different angles
- Engineering Design: Calculate landing distances for projectiles in mechanical systems
- Ballistics: Estimate range for bullets, arrows, or other projectiles
- Safety Planning: Determine safe distances for fireworks, rockets, or launched objects
- Educational Purposes: Understand how launch angle and velocity affect range
Why Use Our Calculator?
- ✅ Quick Calculation: Instantly compute range from velocity and angle without manual trigonometry
- ✅ Shows Maximum Range: Displays the maximum possible range at 45° for comparison
- ✅ Accurate Formula: Uses the precise projectile range equation R = (v₀² sin(2θ)) / g
- ✅ Customizable Gravity: Adjust gravitational acceleration for different planets or scenarios
- ✅ Educational Value: Shows formulas and helps understand angle effects on range
- ✅ Simple Interface: Easy to use with just velocity and angle inputs
Common Applications
Sports Physics: Calculate how far a football, baseball, or golf ball will travel when kicked or hit at different angles, helping athletes optimize their technique.
Engineering Projects: Determine landing distances for projectiles in mechanical launching systems, automated throwing devices, or safety barrier assessments.
Physics Education: Help students understand how launch angle affects range, demonstrating that 45° gives maximum range and complementary angles (30° and 60°) give the same range.
Safety Planning: Calculate landing distances for fireworks, rockets, or other projectiles to ensure safe launch distances and clearances.
Tips for Best Results
- Optimal Angle: For maximum range (from ground level), use 45° launch angle
- Complementary Angles: Angles that sum to 90° (e.g., 30° and 60°) give the same range
- Use Consistent Units: Ensure velocity is in m/s and gravity is in m/s²
- Ground Level Assumption: This formula assumes launch from ground level (h₀ = 0)
- No Air Resistance: Results assume no air resistance; real-world ranges will be shorter
Frequently Asked Questions
What is the maximum range for a given initial velocity?
The maximum range occurs at a 45° launch angle and is given by R_max = v₀² / g. For example, with v₀ = 30 m/s and g = 9.81 m/s², R_max = 900 / 9.81 = 91.74 m. This is the farthest horizontal distance possible for that initial velocity when launched from ground level.
Why do 30° and 60° give the same range?
Because the range formula uses sin(2θ), and sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°). This means complementary angles (angles that sum to 90°) produce the same range. The 30° launch has less height but the same horizontal distance as the 60° launch.
How does initial velocity affect range?
Range is proportional to the square of initial velocity: R ∝ v₀². This means doubling the velocity quadruples the range. For example, if v₀ = 20 m/s gives R = 40 m, then v₀ = 40 m/s gives R = 160 m (4 times the range).
What happens if I launch from an elevated position?
This calculator assumes launch from ground level (h₀ = 0). For elevated launches, the range formula is more complex and depends on initial height. Elevated launches typically have greater range because the projectile has more time in the air before hitting the ground.
Does this calculator account for air resistance?
No, this calculator assumes no air resistance, which is valid for low speeds and dense objects. For high-speed projectiles or light objects, air resistance significantly reduces range. The actual range will be shorter than calculated, especially for high velocities.
Can I use this for projectiles on other planets?
Yes! Simply change the gravity value. For example, on the Moon (g = 1.62 m/s²), ranges will be much longer. On Jupiter (g = 24.79 m/s²), ranges will be much shorter. The formula R = v₀² / g shows that lower gravity increases range.