Solve motion paths for launches in two dimensions. Choose ideal, linear drag, or quadratic drag. See graphs, tables, and landing details in seconds today.
The solver integrates the 2D equations of motion using a fourth-order Runge-Kutta (RK4) method. The state is [x, y, vx, vy].
Drag depends on the selected model, using the relative velocity to wind vrel = (vx-u, vy).
Landing is detected when height crosses zero. The solver interpolates the final step to estimate impact time and range.
For stable results, start with dt = 0.01. If the curve looks jagged, reduce dt.
| Case | Model | v0 | Angle | y0 | Wind | Range (approx.) | Max height (approx.) |
|---|---|---|---|---|---|---|---|
| A | Ideal | 50 m/s | 45 deg | 0 m | 0 m/s | ~255 m | ~64 m |
| B | Linear drag | 50 m/s | 45 deg | 0 m | 0 m/s | Lower than A | Lower than A |
| C | Quadratic drag | 70 m/s | 35 deg | 10 m | 5 m/s | Model dependent | Model dependent |
Rows B and C depend strongly on drag inputs, mass, and step size.
This calculator predicts a two-dimensional flight path from launch speed, launch angle, and starting height. It returns time of flight, range, maximum height, apex location, impact speed, and impact angle. A full time-stamped table is generated for plotting or reporting.
With no drag, acceleration is constant and equal to gravity. A common reference value is 9.80665 m/s² for standard Earth gravity. Ideal motion is useful for quick checks, classroom problems, and sanity tests before enabling drag and wind.
Linear drag is proportional to relative velocity. It is a practical approximation for slow motion in viscous regimes or for simple damping experiments. Increasing the coefficient b reduces range and peak height, and it shifts the apex earlier in time compared with the ideal case.
Quadratic drag scales with speed squared and is common for objects moving through air at moderate to high speeds. The model uses 0.5·ρ·Cd·A, where ρ is air density, Cd is drag coefficient, and A is cross-sectional area. Typical sea-level air density is about 1.225 kg/m³.
Wind is treated as a horizontal flow that changes the relative speed seen by the projectile. A tailwind effectively lowers drag and can extend range, while a headwind increases drag and shortens the trajectory. Because the drag force uses relative velocity, even small winds can matter at low launch speeds.
The trajectory is integrated with a fourth-order Runge–Kutta method. Accuracy depends on the time step dt. Values like 0.01 s are usually smooth for typical sports-scale launches, while faster motion or stronger drag can benefit from 0.005 s or smaller. Very small dt increases computation time and table size.
Landing is detected when the height crosses zero. The calculator interpolates within the last step to estimate impact time and range more precisely than a raw step boundary. If the projectile does not land within the selected max time, the solver still reports the end-state and flags the run.
Start with the ideal model to confirm inputs and units. Then enable drag and use realistic parameters: for a sphere, Cd near 0.47 is a common starting point, while area A follows from diameter. Compare CSV exports across scenarios to quantify sensitivity, and keep dt consistent when comparing runs.
Drag continuously removes kinetic energy, lowering both horizontal and vertical speeds. The projectile slows earlier, reaches a smaller apex, and spends less time traveling forward. Quadratic drag can be especially strong at high speeds.
Use 0.01 s for many general cases. If you see jagged curves, unstable landing, or strong drag effects, try 0.005 s. Smaller steps improve accuracy but increase runtime and the number of table rows.
For a circular object, A = π(d/2)² using its diameter d in meters. For non-circular shapes, use the projected frontal area facing the airflow. A larger area increases drag and reduces range.
Wind is a constant horizontal air velocity. Drag uses the projectile velocity relative to that wind. A positive wind value acts like a tailwind; negative values act like a headwind and increase deceleration.
Yes. Set the initial height to your platform height. The solver detects when the trajectory crosses ground level and reports time of flight, range, and impact conditions from that elevated start.
This happens if max time is too small, if the object never falls below y = 0 in the simulated window, or if numerical settings are too coarse. Increase max time or reduce dt to improve detection.
The reported angle comes from atan2(vy, vx). It is typically negative at impact because vy is downward. A more negative value indicates a steeper downward approach at landing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.