Last Updated: October 20, 2025
Calculate projectile trajectory instantly with our advanced 2025 physics calculator to analyze projectile paths and motion characteristics for educational and professional applications.
Enter the initial velocity, launch angle, and time values in your preferred units. The calculator will automatically convert between units and display the trajectory coordinates in multiple formats. Results update instantly as you type.
Enter values to see results
The Trajectory Calculator is a specialized tool that calculates the path followed by a projectile through space. This fundamental concept is essential for understanding projectile motion and ballistics.
For more information about trajectory and projectile motion, visit Illustrations: Trajectory and Wikipedia: Projectile Motion.
In physics, trajectory describes the curved path of a projectile under the influence of gravity. The path follows a parabolic shape determined by the initial velocity, launch angle, and gravitational acceleration. This principle is fundamental in understanding ballistics and is essential for range calculations and maximum height analysis.
Trajectory is the curved path followed by a projectile through space under the influence of gravity.
Whether you're studying physics, analyzing ballistics, understanding sports trajectories, or solving problems involving projectile motion, this calculator provides accurate, instant results with flexible unit conversions to meet your specific needs. For related calculations, explore our projectile motion calculator, time of flight calculator, projectile range calculator, maximum height calculator, and muzzle velocity calculator.
x = v₀cos(θ)t
y = v₀sin(θ)t - ½gt²
These parametric equations calculate trajectory coordinates using initial velocity, launch angle, time, and gravity.
Initial velocity (v₀) is the speed at which the projectile is launched. Higher initial velocities result in longer trajectories because the projectile travels further before gravity brings it down.
Launch angle (θ) is the angle at which the projectile is launched relative to the horizontal. The optimal angle for maximum range is 45°, but different angles create different trajectory shapes.
Key trajectory characteristics:
The trajectory calculation is crucial for understanding projectile motion and ballistics. It helps determine the path a projectile will follow and is essential for calculating range, maximum height, and impact points.
| Field/Application | Typical Range | Importance |
|---|---|---|
| Sports Ballistics | 10-100 meters | Critical for performance analysis and strategy |
| Military Ballistics | 100-50000 meters | Essential for targeting and accuracy |
| Space Missions | 1000-400000 km | Critical for orbital mechanics and missions |
| Water Sports | 5-50 meters | Important for safety and performance |
| Fireworks Displays | 50-500 meters | Essential for timing and safety |
| Agricultural Spraying | 1-20 meters | Critical for coverage and efficiency |
| Educational Physics | 1-50 meters | Fundamental for understanding motion principles |
| Entertainment Industry | 10-1000 meters | Important for special effects and stunts |
Given:
Step 1: Apply trajectory formulas
x = v₀cos(θ)t
y = v₀sin(θ)t - ½gt²
Step 2: Calculate horizontal position
x = 20 × cos(30°) × 1.0 = 20 × 0.866 × 1.0 = 17.32 m
Step 3: Calculate vertical position
y = 20 × sin(30°) × 1.0 - ½ × 9.81 × 1.0²
y = 20 × 0.5 × 1.0 - 4.905 = 10 - 4.905 = 5.095 m
Final Answer
(17.32 m, 5.095 m)
Trajectory coordinates
Given:
Step 1: Apply trajectory formulas
x = v₀cos(θ)t
y = v₀sin(θ)t - ½gt²
Step 2: Calculate horizontal position
x = 100 × cos(45°) × 2.0 = 100 × 0.707 × 2.0 = 141.4 m
Step 3: Calculate vertical position
y = 100 × sin(45°) × 2.0 - ½ × 9.81 × 2.0²
y = 100 × 0.707 × 2.0 - 19.62 = 141.4 - 19.62 = 121.78 m
Final Answer
(141.4 m, 121.78 m)
Trajectory coordinates
Given:
Step 1: Apply trajectory formulas
x = v₀cos(θ)t
y = v₀sin(θ)t - ½gt²
Step 2: Calculate horizontal position
x = 8 × cos(60°) × 0.5 = 8 × 0.5 × 0.5 = 2.0 m
Step 3: Calculate vertical position
y = 8 × sin(60°) × 0.5 - ½ × 9.81 × 0.5²
y = 8 × 0.866 × 0.5 - 1.226 = 3.464 - 1.226 = 2.238 m
Final Answer
(2.0 m, 2.238 m)
Trajectory coordinates
💡 Did you know? The longest recorded trajectory for a human projectile (skydiver) covered over 39 km horizontally!
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