This calculator fits Contracts & Documents workflows that need report-ready motion estimates, exportable tables, and clear supporting calculations for technical exhibits, review notes, and project records.
Calculator Input
Example Data Table
| Case | Initial speed (m/s) | Angle (°) | Initial height (m) | Target height (m) | Final speed (m/s) | Time (s) | Distance (m) |
|---|---|---|---|---|---|---|---|
| Short launch | 20.00 | 35.00 | 1.00 | 0.00 | 20.48 | 2.42 | 39.69 |
| High arc | 28.00 | 55.00 | 1.50 | 0.00 | 28.52 | 4.74 | 76.13 |
| Elevated target | 32.00 | 50.00 | 0.00 | 8.00 | 29.45 | 4.65 | 95.58 |
| Low angle | 40.00 | 20.00 | 2.00 | 0.00 | 40.49 | 2.93 | 110.07 |
| Tall release | 22.00 | 42.00 | 12.00 | 0.00 | 26.82 | 3.67 | 59.97 |
Formula Used
Horizontal velocity: vx = v0 cos(θ)
Initial vertical velocity: vy0 = v0 sin(θ)
Vertical position: y(t) = y0 + vy0t - 0.5gt²
Impact time at target height: solve y(t) = yf
Final speed: vf = √(vx² + vyf²)
Energy shortcut: vf = √(v0² + 2g(y0 - yf))
This shortcut confirms the magnitude of final velocity when air resistance is ignored. The component method then adds direction, travel time, and horizontal distance.
The calculator chooses a physically valid time root based on the selected branch. Use ascending when the projectile reaches the target height while rising. Use descending for impact-style cases. Auto favors the descending solution when one exists.
How to Use This Calculator
- Choose metric or imperial units first.
- Enter the initial speed and launch angle.
- Add the starting height and desired target height.
- Keep standard gravity or enter a custom value.
- Select auto, ascending, or descending branch behavior.
- Pick the decimal precision for displayed results.
- Press the calculate button to show results above the form.
- Review the graph, summary values, and export files if needed.
Frequently Asked Questions
1. What does final velocity mean here?
It is the projectile velocity at the chosen target height. The tool reports total speed, vertical component, horizontal component, direction angle, travel time, and horizontal distance at that moment.
2. Why can the same target height have two solutions?
A projectile can pass the same height once while rising and again while falling. The branch selector lets you choose which physically meaningful point you want to report.
3. Why does final speed sometimes ignore launch angle?
Without air resistance, the final speed magnitude depends on starting speed and height difference through energy conservation. The launch angle still changes time, range, and vertical direction.
4. What causes an unreachable target height error?
The entered speed and angle may not provide enough vertical reach to hit the specified height. Increase the launch speed, adjust the angle, or lower the target height.
5. Does this calculator include drag or wind?
No. It assumes ideal projectile motion with constant gravity and no air resistance. That keeps the equations clean and makes the exports easier to verify in technical documentation.
6. Can I use imperial units?
Yes. Select imperial units and enter speed in feet per second, heights in feet, and gravity in feet per second squared. Keep all values within one consistent system.
7. What is the graph showing?
The Plotly chart shows horizontal distance on the x-axis and height on the y-axis. It plots the projectile path from launch until the chosen target height is reached.
8. What do the CSV and PDF downloads contain?
They include the input values, calculated summary metrics, and sample trajectory points. This helps you attach clear calculation evidence to reports, exhibits, and review packages.