Trajectory Optimization Tool

Calculator Inputs

The page stays in a single-column flow, while the input grid adapts to three columns on large screens, two on smaller screens, and one on mobile.

Launch height (m)

Starting vertical position.

Target distance X (m)

Horizontal distance to the target.

Target height Y (m)

Vertical height to reach.

Gravity (m/s²)

Use 9.81 for Earth.

Minimum speed (m/s)

Lower search boundary.

Maximum speed (m/s)

Upper search boundary.

Speed step

Smaller steps increase precision.

Minimum angle (degrees)

Keep above 0 degrees.

Maximum angle (degrees)

Keep below 89.9 degrees.

Angle step

Smaller steps search more paths.

Hit tolerance (m)

Defines exact-hit acceptance.

Accuracy weight

Higher values prioritize smaller miss distance.

Energy weight

Higher values prefer lower launch speed.

Time weight

Higher values prefer faster arrival.

Obstacle distance X (m)

Use 0 to disable obstacle checking.

Obstacle height (m)

Only used when obstacle distance is active.

Required clearance (m)

Extra height above the obstacle.

Example Data Table

Scenario	Target X	Target Y	Speed Range	Angle Range	Obstacle	Best Angle	Best Speed	Miss	Time
Sample optimization run	120 m	20 m	20 to 60 m/s	10° to 80°	60 m, 12 m high	31.5°	38.0 m/s	0.18 m	3.72 s

Formula Used

This tool searches a range of launch angles and speeds, then ranks each valid path by a weighted optimization score.

Horizontal motion: x(t) = v cos(θ) · t

Vertical motion: y(t) = y₀ + v sin(θ) · t − 0.5 g t²

Time to target: t_target = X_target / (v cos(θ))

Height at target: y_target = y₀ + v sin(θ)t_target − 0.5 g t_target²

Miss distance: miss = | y_target − Y_target |

Apex height: apex = y₀ + (v² sin²(θ)) / (2g)

Total flight time: t_flight = [v sin(θ) + √(v² sin²(θ) + 2gy₀)] / g

Optimization score: score = w₁·normalized miss + w₂·normalized energy + w₃·normalized time

The tool rejects paths that cannot reach the target distance before landing. It can also reject paths that fail the obstacle and clearance requirement.

How to Use This Calculator

Enter the target distance and target height.
Set the launch height and gravity value.
Choose minimum and maximum speeds for the search.
Choose minimum and maximum angles for the search.
Use smaller step sizes for finer optimization.
Adjust the accuracy, energy, and time weights.
Add obstacle values if a clearance check is needed.
Click Optimize Trajectory to view the best result.
Review the summary cards, top solution table, and chart.
Export the results with the CSV or PDF buttons.

FAQs

1. What does this tool optimize?

It searches combinations of speed and launch angle, then ranks valid paths using weighted accuracy, energy, and time criteria. The best result may be an exact hit or the closest feasible approximation.

2. Why does the best result sometimes miss the target?

If no tested angle and speed pair lands inside the selected tolerance, the tool returns the best approximate path. Reducing step size or widening the search range usually improves the fit.

3. What does the tolerance value do?

Tolerance defines how close a path must be to count as a direct hit. A smaller tolerance is stricter, while a larger tolerance marks more solutions as acceptable.

4. How are the weights interpreted?

Accuracy weight favors smaller miss distance. Energy weight favors lower launch speed. Time weight favors shorter travel time. The weights are normalized automatically before scoring.

5. Does the tool include air resistance?

No. This version uses classical projectile equations without aerodynamic drag. It is best for mathematical exploration, planning, estimation, and comparison under simplified motion assumptions.

6. What happens when I add an obstacle?

The tool checks each path at the obstacle distance. A trajectory only remains valid if it clears the obstacle height plus the required clearance margin.

7. Why do smaller step values take longer?

Smaller angle and speed increments create more candidate combinations. That improves precision, but it also increases the number of evaluations and therefore the processing time.

8. What is included in the exports?

The CSV export includes the current inputs, best solution summary, and ranked solution table. The PDF export provides a concise report with the main optimization metrics and top paths.