Inputs
Example Data Table
| Distance (m) | Speed (m/s) | Launch height (m) | Target height (m) | Gravity (m/s²) | Typical angle results (deg) |
|---|---|---|---|---|---|
| 60 | 28 | 0 | 0 | 9.81 | ≈ 22.06° and 67.94° |
| 45 | 24 | 1.5 | 0 | 9.81 | Two solutions if reachable |
| 30 | 18 | 0 | 2 | 9.81 | Higher arc may be required |
Formula Used
The vertical position of a projectile (no drag) can be written as:
y = y0 + x·tan(θ) − g·x² / (2·v0²·cos²(θ))
For a target at horizontal distance R and height yT, set x = R and y = yT.
Let Δy = yT − y0 and T = tan(θ). This becomes a quadratic:
A·T² − R·T + (A + Δy) = 0, where A = g·R² / (2·v0²).
Solve for T, then compute θ = arctan(T). The discriminant must be non‑negative for real solutions.
How to Use This Calculator
- Enter the horizontal distance from launch to the target.
- Enter the initial speed and choose the correct unit.
- Set launch height and target height for elevation changes.
- Use standard gravity, or enter a custom value.
- Press Solve Launch Angle to view solutions above.
- Download a CSV for logs, or save a PDF printout.
Projectile Launch Angle Guide
1) What the solver finds
This tool computes the launch angle θ that allows a projectile with initial speed v0 to reach a target located a horizontal distance R away. If the target height differs from the launch height, the solver also accounts for Δy = yT − y0. When conditions permit, two angles exist: a low, fast trajectory and a high, arcing trajectory.
2) Key inputs and units
Range can be entered in meters, centimeters, millimeters, kilometers, feet, inches, or yards. Speed supports m/s, km/h, mph, and ft/s. Gravity supports m/s² and ft/s². All inputs are converted internally to consistent base units before calculation, then converted back for display and downloads.
3) The reachability test
The core equation becomes a quadratic in T = tan(θ) with parameter A = gR²/(2v0²). A real solution requires a non‑negative discriminant: D = R² − 4A(A + Δy). If D is negative, the requested target cannot be reached with the chosen speed, distance, and height difference.
4) Low angle vs high angle behavior
The low‑angle solution typically has shorter flight time and a lower peak height, which can reduce exposure to cross‑winds. The high‑angle solution increases time of flight and peak height, which may help clear obstacles but can raise impact angle and increase sensitivity to disturbances.
5) Derived outputs you can use
For each valid angle, the calculator reports flight time t, horizontal speed vx = v0 cos(θ), initial vertical speed vy0 = v0 sin(θ), vertical speed at the target, peak height hmax, and impact speed. These values support planning for clearance, timing, and expected arrival conditions.
6) Practical engineering and lab scenarios
Typical uses include verifying launcher settings, setting up range experiments, sports ballistics approximations, and educational demonstrations. For equal launch and target heights, two complementary angles often appear; their sum is near 90° in ideal conditions when solutions exist.
7) Sensitivity and uncertainty
Small changes in v0 or R can shift the angles noticeably, especially when operating near the reachability limit where D approaches zero. If your inputs come from measurements, use the rounding control and rerun with slightly varied values to understand uncertainty bands.
8) Assumptions and limitations
Results assume uniform gravity, flat Earth over the range, and no air drag or lift. At high speeds, long distances, or for objects with significant aerodynamic effects, real trajectories may deviate. Use this solver for first‑order estimates and validation before applying higher‑fidelity models.
FAQs
1) Why do I sometimes get two angles?
For many targets, both a low arc and a high arc can land at the same range. They differ in flight time and peak height, even with identical starting speed.
2) What does “no real launch angle” mean?
The inputs fail the reachability condition, so the quadratic has no real solution. Increase speed, reduce distance, or adjust target and launch heights to restore feasibility.
3) Which angle should I choose for practice?
Choose the lower angle for shorter flight time and lower peak height. Choose the higher angle to clear obstacles, accepting longer time of flight and more sensitivity.
4) Does the solver include air resistance?
No. It assumes ideal projectile motion with constant gravity and no drag. If drag is significant, the required angle typically increases and the range decreases.
5) What if the target is higher than the launch?
A positive height change requires more vertical component to reach the target. Depending on speed and distance, you may get one angle, two angles, or no solution.
6) Why are my results different from a real test?
Measurement error, launch angle calibration, wind, spin, and aerodynamic lift or drag can shift the landing point. Use this tool as a baseline, then refine with empirical corrections.
7) How do CSV and PDF outputs help?
CSV supports quick logging, plotting, and reporting in spreadsheets. The PDF option uses your browser print dialog to save a clean report of inputs and computed solutions.