Calculator Form
Formula Used
This calculator uses standard projectile motion equations with constant gravity and no air resistance. The motion is modeled with separate horizontal and vertical equations.
| Horizontal position | x(t) = x₀ + vₓt |
|---|---|
| Vertical position | y(t) = y₀ + vᵧt - (1/2)gt² |
| Velocity components from speed and angle | vₓ = v₀ cos(θ), vᵧ = v₀ sin(θ) |
| Time to apex | t_apex = vᵧ / g when vᵧ > 0 |
| Maximum height | y_max = y₀ + vᵧ² / (2g) when vᵧ > 0 |
| Ground intersection time | Solve y₀ + vᵧt - (1/2)gt² = 0 and keep the largest non-negative real root. |
| Arc length | The path length is approximated numerically by summing distances between consecutive plotted points. |
How to Use This Calculator
- Choose either Initial speed + launch angle or Horizontal and vertical velocity components.
- Enter the starting coordinates, gravity value, and time step.
- Select degrees or radians when you use angle-based input.
- Optionally enter a custom end time to limit the plotted duration.
- Keep the ground-stop option checked to end the graph at landing, when a real landing time exists.
- Click Plot Trajectory to calculate the metrics and draw the Plotly graph.
- Review the summary table, plotted sample points, and computed equations.
- Use the CSV and PDF buttons in the results section to save the current output.
Example Data Table
Example scenario using speed-angle input with a launch speed of 30 m/s, angle 40°, initial height 1.5 m, gravity 9.81 m/s², and time step 0.1 s.
| Item | Value |
|---|---|
| Initial x-coordinate | 0.0000 m |
| Initial y-coordinate | 1.5000 m |
| Initial speed | 30.0000 m/s |
| Launch angle | 40.0000° |
| Initial horizontal velocity | 22.9813 m/s |
| Initial vertical velocity | 19.2836 m/s |
| Time of flight | 4.0077 s |
| Maximum height | 20.4530 m |
| Horizontal range | 92.1029 m |
| Apex time | 1.9657 s |
Frequently Asked Questions
1. What does this trajectory plotter calculate?
It calculates the plotted path, time of flight, apex time, maximum height, landing position, landing speed, final plotted point, and approximate arc length for a two-dimensional projectile.
2. Can I enter the angle in radians?
Yes. Choose radians from the angle unit menu before entering the launch angle. The calculator also displays the computed launch angle using the currently selected unit.
3. What is the difference between the two input modes?
The first mode starts from speed and angle. The second uses direct horizontal and vertical velocity components. Both modes lead to the same equations after the velocity components are known.
4. Why might the graph stop before the custom end time?
If the ground-stop option is enabled and the motion reaches ground level earlier, the plot ends at landing. Uncheck that option to keep plotting until your chosen end time.
5. Does this calculator include air resistance?
No. This page uses ideal projectile motion with constant gravitational acceleration and no drag. That keeps the equations simple and the graph fast to calculate and export.
6. Why is arc length different from horizontal range?
Horizontal range measures only the x-direction distance from launch to landing. Arc length measures the actual curved path, so it is usually larger than the range.
7. Can I export the plotted results?
Yes. Use the CSV button to download the summary and all plotted points. Use the PDF button to save the visible results section as a PDF document.
8. What happens if the trajectory never reaches ground level?
When there is no real non-negative ground intersection, the landing metrics are shown as unavailable. The graph then uses your custom end time or an automatic fallback duration.