🎯 Trajectory Calculator
Calculate projectile trajectory
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 at launch. Higher initial velocity results in longer range and higher maximum height.
Enter Launch Angle
Input the angle at which the projectile is launched above the horizontal, measured in degrees. This is crucial: 45° gives maximum range for ground-level launches, while 90° gives maximum height but zero range. Typical angles range from 0° (horizontal) to 90° (vertical).
Enter Initial Height (Optional)
If the projectile starts from a height above ground level, enter that initial height in meters. Leave as 0 if launched from ground level. Higher initial height increases both range and time of flight. Useful for projectiles launched from towers, cliffs, or elevated platforms.
Set Gravity (Optional)
The default gravity is 9.81 m/s² (Earth's gravity). You can change this value for calculations on other planets or in different gravitational fields. Lower gravity means longer flight times and greater ranges.
Click Calculate
Press the "Calculate" button to compute the trajectory parameters: range (horizontal distance), maximum height, time of flight, and time to reach maximum height. All values are displayed in the results section.
Formula
R = (v₀² sin(2θ)) / g
H_max = h₀ + (v₀² sin²(θ)) / (2g)
T = (v₀ sin(θ) + √(v₀² sin²(θ) + 2gh₀)) / g
Where:
R = Range (horizontal distance, m)
H_max = Maximum height (m)
T = Time of flight (s)
vâ‚€ = Initial velocity (m/s)
θ = Launch angle (degrees)
hâ‚€ = Initial height (m)
g = Gravitational acceleration (9.81 m/s²)
Note: Range formula assumes ground-level launch (h₀ = 0). For elevated launches, use T × v₀ cos(θ).
Example 1: Projectile at 45° from ground
Given: v₀ = 30 m/s, θ = 45°, h₀ = 0 m, g = 9.81 m/s²
Step 1: Calculate range
R = (30² × sin(90°)) / 9.81 = (900 × 1) / 9.81 = 91.74 m
Step 2: Calculate maximum height
H_max = 0 + (30² × sin²(45°)) / (2 × 9.81)
H_max = (900 × 0.5) / 19.62 = 22.93 m
Step 3: Calculate time of flight
T = (30 × sin(45°) + √(30² × sin²(45°) + 0)) / 9.81
T = (21.21 + 21.21) / 9.81 = 4.32 s
Example 2: Projectile from elevated position
Given: v₀ = 25 m/s, θ = 60°, h₀ = 10 m, g = 9.81 m/s²
v₀y = 25 × sin(60°) = 21.65 m/s
H_max = 10 + (21.65²) / (2 × 9.81) = 10 + 23.89 = 33.89 m
T = (21.65 + √(21.65² + 2×9.81×10)) / 9.81 = 4.84 s
R = v₀ cos(θ) × T = 25 × cos(60°) × 4.84 = 12.5 × 4.84 = 60.5 m
Understanding Trajectory Parameters
• Maximum range occurs at 45° for ground-level launches
• Maximum height occurs at 90° (vertical launch), but range is zero
• Range = 0 at both 0° and 90° (horizontal and vertical launches)
• Time of flight increases with launch angle and initial height
• Higher initial velocity increases range, height, and flight time
• For symmetric trajectories (h₀ = 0), time to max height = T/2
About Trajectory Calculator
The Trajectory Calculator is a comprehensive physics tool for analyzing projectile motion - the curved path followed by objects launched into the air. It calculates key trajectory parameters including range (horizontal distance traveled), maximum height reached, time of flight (total time in the air), and time to reach maximum height. The calculator uses standard projectile motion equations derived from kinematics, accounting for initial velocity, launch angle, initial height, and gravitational acceleration. Understanding trajectories is essential for physics, engineering, sports analysis, ballistics, and any application involving launched objects.
When to Use This Calculator
- Physics Problems: Solve projectile motion problems involving range, height, and flight time
- Sports Analysis: Calculate trajectories for balls in sports like baseball, basketball, football, or golf
- Engineering Design: Analyze projectile trajectories for launching systems, catapults, or automated throwing devices
- Ballistics: Calculate trajectories for bullets, arrows, or other projectiles
- Safety Planning: Determine landing points and maximum heights for fireworks, rockets, or projectiles
- Educational Purposes: Understand how launch angle and velocity affect trajectory parameters
Why Use Our Calculator?
- ✅ Comprehensive Results: Calculates range, maximum height, time of flight, and time to max height
- ✅ Accounts for Initial Height: Handles projectiles launched from elevated positions
- ✅ Physics-Based Formulas: Uses standard projectile motion equations from kinematics
- ✅ Educational Value: Shows formulas and step-by-step calculations for learning
- ✅ Flexible Input: Adjust gravity for different planets or scenarios
- ✅ Instant Analysis: Get all trajectory parameters quickly without manual computation
Common Applications
Sports Physics: Analyze trajectories for balls in baseball (home runs, fly balls), basketball (free throws, three-pointers), football (punts, field goals), or golf (drives, approach shots). Understand optimal launch angles for maximum distance.
Physics Education: Help students understand projectile motion, the effects of launch angle and velocity, and how to apply kinematic equations to real-world scenarios. Demonstrates why 45° gives maximum range for ground-level launches.
Engineering Projects: Design launching systems, calculate trajectories for automated throwing devices, or analyze projectile motion in mechanical systems. Essential for robotics and automation applications.
Safety and Ballistics: Calculate landing points and maximum heights for fireworks, rockets, or projectiles. Important for safety planning and determining safe launch distances.
Tips for Best Results
- Optimal Angle: For maximum range from ground level, use 45°. For other initial heights, optimal angle varies
- Initial Height: Don't forget to include initial height if launching from elevated position; it significantly affects range and flight time
- No Air Resistance: This calculator assumes no air resistance; real-world trajectories will differ, especially at high speeds
- Angle Range: Launch angles should be between 0° (horizontal) and 90° (vertical) for meaningful results
- Units Consistency: Ensure all inputs use consistent units (meters, seconds, m/s, m/s²)
Frequently Asked Questions
What angle gives maximum range?
For projectiles launched from ground level (h₀ = 0), 45° gives maximum range. This is because the range formula R = (v₀² sin(2θ)) / g is maximum when sin(2θ) = 1, which occurs at 2θ = 90°, so θ = 45°. For elevated launches, the optimal angle depends on initial height.
How does initial height affect the trajectory?
Initial height increases both range and time of flight. The projectile has more time to fall, so it travels further horizontally. Maximum height also increases, but by a smaller amount. The effect is most significant for longer flight times.
Why is the range zero at 0° and 90°?
At 0° (horizontal launch), the projectile has no vertical component initially and immediately starts falling, hitting the ground quickly with minimal horizontal travel. At 90° (vertical launch), the projectile goes straight up and down, with zero horizontal displacement, so range is zero.
Does this calculator account for air resistance?
No, this calculator assumes no air resistance (ideal projectile motion). In reality, air resistance reduces range, maximum height, and flight time, especially at high speeds. For accurate real-world trajectories, air resistance must be considered using numerical methods.
Can I use this for projectiles on other planets?
Yes! Simply change the gravity value. For example, use g = 1.62 m/s² for the Moon (longer ranges and flight times), g = 3.71 m/s² for Mars, or g = 24.79 m/s² for Jupiter (shorter ranges and flight times). Lower gravity means longer trajectories.
What's the relationship between range and maximum height?
For ground-level launches at 45°, maximum height equals one-quarter of the range. Generally, range and maximum height both increase with initial velocity, but their relationship depends on launch angle. Maximum height increases with angle, while range peaks at 45°.