Calculation Results
Trajectory Graph
Selected Point Analysis
Step-by-Step Calculation
Trajectory Table
| Time | Distance | Height | Horizontal Velocity | Vertical Velocity | Total Velocity | Kinetic Energy | Path Angle |
|---|
Formula Used
Ideal motion separates horizontal and vertical movement. Gravity changes only vertical velocity. Drag adds a velocity-dependent resisting force to both directions.
x(t) = v₀ cos(θ)t
y(t) = h₀ + v₀ sin(θ)t − ½gt²
H = h₀ + v₀²sin²(θ) / (2g)
Fd = ½ρCdAv²
KE = ½mv²
How to Use This Calculator
- Select an ideal or drag-based trajectory model.
- Enter velocity, angle, elevations, and projectile properties.
- Choose environmental conditions and numerical precision.
- Set simulation limits and an optional comparison scenario.
- Calculate, inspect the graph, and export your results.
Example Data
| Scenario | Velocity | Angle | Mass | Diameter | Model |
|---|---|---|---|---|---|
| Classroom sphere | 50 m/s | 45° | 145 g | 73 mm | Drag |
| Foam training ball | 22 m/s | 38° | 30 g | 90 mm | Drag |
| Ideal comparison | 40 m/s | 50° | 100 g | 60 mm | Ideal |
Assumptions and Limitations
The ideal model ignores air resistance and wind. The drag model uses a constant drag coefficient. Real shapes may require advanced aerodynamic data and testing.
The atmosphere estimate uses simplified pressure and humidity equations. It cannot reproduce every local condition. Treat all outputs as educational simulation estimates only.
Frequently Asked Questions
What does the ideal model calculate?
It uses standard projectile equations without air resistance. Horizontal velocity stays constant during flight. Vertical velocity changes because gravity acts downward continuously.
How does drag change the result?
Drag reduces horizontal and vertical speed over time. It usually lowers range and maximum height. The effect grows rapidly as projectile speed increases.
Which numerical method is most accurate?
Runge–Kutta is usually the strongest available choice. Improved Euler offers a useful middle option. Smaller time steps generally improve numerical accuracy further.
Why does projectile diameter matter?
Diameter determines the estimated frontal area. A larger area produces stronger aerodynamic drag. Shape still matters through the drag coefficient value.
Can landing elevation differ from launch elevation?
Yes, both elevations can be entered independently. The calculator detects the selected landing level. This supports raised or lowered educational launch scenarios.
How is air density estimated?
Temperature, pressure, humidity, and altitude are combined. Moist air uses separate dry-air and vapor terms. The result remains a simplified atmospheric estimate.
What happens when a simulation limit is reached?
The calculator stops and displays a warning. Increase time or distance limits carefully. Very large limits may create slower calculations.
Can I compare two trajectories?
Enable comparison and enter another velocity and angle. Both paths appear on the same graph. Summary values remain based on the primary scenario.
Are these results suitable for targeting?
No, this tool is for education and simulation. It excludes operational targeting and weapon-specific models. Never rely on it for harmful activities.