Launch angle
Angle changes both components at once. Near vertical launch, a small angle change can create a large relative change in horizontal range even when speed barely changes.
Physics and mechanics
Projectile Conversion Calculator
Convert projectile launch speed and calculated range between SI and US customary units while preserving vector components, launch height, gravity, and unrounded flight time.
PROJECTILE UNIT BRIDGE
Convert the launch state before solving the trajectory
Unit conversion is not a cosmetic step in projectile work. This calculator converts the entered launch speed to a single SI basis, resolves horizontal and vertical components, solves the positive landing-time root, and then reports range in metres and feet.
Speed in m/s-
Speed in ft/s-
Horizontal velocity-
Vertical velocity-
Flight time-
Range in metres-
Range in feet-
Maximum height-
PROJECTILE UNIT BRIDGE
Conversion and trajectory reconciliation
All trajectory calculations use metres, seconds, and the entered SI gravity. US customary outputs are converted only after the unrounded SI solution is complete.
| Quantity | Source unit | Source value | Target unit | Target value |
|---|
DETAILED CALCULATION PROCESS
SI-first projectile solution: formula, units, substitution, and reconciliation
1. Start from the governing relation
v = vkm/h / 3.6; vx = v cos(theta); vy = v sin(theta); t = (vy + sqrt(vy² + 2gh0))/g; R = vx t
Convert speed to m/s, resolve components using degrees converted to radians, solve y(t)=h0+vyt-gt²/2 for the positive time, and convert the resulting range with 1 m = 3.280839895 ft.
2. Define every symbol before substituting numbers
| Symbol | Meaning | Unit | Default-page basis |
|---|---|---|---|
| v | Launch speed after conversion | m/s | km/h ÷ 3.6 |
| theta | Launch angle above horizontal | degrees | entered angle |
| vx | Horizontal velocity component | m/s | v cos(theta) |
| vy | Vertical velocity component | m/s | v sin(theta) |
| t | Positive flight time | s | positive quadratic root |
| R | Horizontal range | m | vx × t |
3. Record the entered assumptions
- Launch speed (km/h): 72. Converted exactly to m/s before component resolution.
- Launch angle above horizontal (degrees): 38. Must remain strictly between 0 and 90 degrees.
- Launch height above landing level (m): 1.8. Vertical offset used in the landing-time quadratic.
- Gravitational acceleration (m/s²): 9.80665. Standard gravity is the default; use a documented local value when needed.
4. Normalize units and conventions
- Convert kilometres per hour to metres per second by dividing by 3.6 before resolving components.
- Trigonometric functions require the entered degree angle to be converted to radians internally.
- Keep height in metres and gravity in metres per second squared so the time root remains dimensionally consistent.
5. Follow the live substitution ledger
6. Reconcile the result before using it
RESULT INTERPRETATION
Read converted components before trusting the range
The horizontal and vertical velocity components show how the entered speed and angle are divided at launch. Flight time is controlled by the vertical component, initial height, and gravity; horizontal range then uses the unchanged horizontal component under the no-drag assumption. A correct unit conversion does not validate that assumption.
Zero launch angle can still produce range when the initial height is positive. A negative or nonphysical flight-time branch is discarded by the model, while an angle near 90 degrees makes horizontal range highly sensitive to rounding. Use the component ledger to distinguish conversion effects from trajectory assumptions.
DECISION BOUNDARY
What the calculated status does and does not decide
All trajectory calculations use metres, seconds, and the entered SI gravity. US customary outputs are converted only after the unrounded SI solution is complete.
SENSITIVITY AND STRESS TESTING
Inputs that dominate the converted trajectory
Initial height
Height extends flight time without increasing horizontal velocity. Measure it from the modeled release point to the landing datum, not from an unrelated site benchmark.
Drag and wind
The closed-form range omits aerodynamic force. For large, light, fast, or irregular projectiles, a numerical drag model can matter more than additional conversion precision.
HOW TO USE THIS CALCULATOR
Convert and solve without mixing unit systems
- Enter launch speed in kilometres per hour and keep the stated angle convention above horizontal.
- Measure launch height from the release point to the landing elevation, not to an arbitrary floor reference.
- Use the gravitational value required by the problem; standard gravity is 9.80665 m/s².
- Read the component ledger before using flight time or range in downstream work.
- Keep unrounded SI outputs for reconciliation and round only the final reported unit.
SUBJECT FOUNDATIONS
Five principles behind projectile conversion
- Scalar speed
- Magnitude of launch velocity before direction is resolved.
- Velocity components
- Orthogonal horizontal and vertical parts created by the launch angle.
- Positive landing root
- The later physically meaningful solution of the vertical-position quadratic.
- Reference elevation
- Launch height is defined relative to the landing level used by the model.
- Conversion sequencing
- Convert inputs to one coherent basis, calculate, then convert outputs.
MODEL BOUNDARY
SI-first projectile solution
v = vkm/h / 3.6; vx = v cos(theta); vy = v sin(theta); t = (vy + sqrt(vy² + 2gh0))/g; R = vx t
Convert speed to m/s, resolve components using degrees converted to radians, solve y(t)=h0+vyt-gt²/2 for the positive time, and convert the resulting range with 1 m = 3.280839895 ft.
DECISION DEPTH
Where conversion mistakes enter the physics
Angles are dimensionless but not format-free
Programming functions usually require radians. Entered degrees must be converted once; converting twice changes both components.
Range depends on launch and landing elevation
The familiar v²sin(2theta)/g expression applies only when launch and landing heights are equal. This page solves the general positive root.
Rounded components can break reconciliation
If vx and vy are rounded before the time calculation, the final range may not agree with the speed and angle. Use the unrounded internal values.
REAL USE CASES
Two conversion-led trajectory checks
Sports launch data in km/h
A launch monitor reports speed in km/h while a coaching report needs ft/s and feet. One SI calculation prevents mixing unit systems inside the trajectory.
Lab launcher above the landing table
A student measures a nonzero release height. The positive-root ledger shows why equal-height range shortcuts do not apply.
TERMS USED ON THIS PAGE
Projectile conversion vocabulary
- Launch speed
- Magnitude of velocity at the chosen initial instant.
- Launch angle
- Direction of velocity measured above horizontal.
- Horizontal component
- v cos(theta), constant only in the no-drag model.
- Vertical component
- v sin(theta), changed by gravity after launch.
- Flight time
- Positive elapsed time until the projectile reaches landing elevation.
- Range
- Horizontal displacement accumulated during flight.
EVIDENCE TO RETAIN
Retain units with every measured value
Record instrument model, calibration or stated accuracy, speed unit, angle convention, launch and landing reference points, gravitational value, environmental conditions, and unrounded converted values. Store the original readings rather than only the final feet or metres.
LIMITS AND EXCLUSIONS
Assumptions in the conversion trajectory
- The model neglects aerodynamic drag, lift, wind, spin, curvature, and changing gravity.
- Launch speed and angle are treated as simultaneous initial conditions at one point.
- The landing surface is represented by one fixed elevation.
- This educational result is not a ballistics, weapons, safety-zone, or engineering certification.
RELIABLE SOURCES
References supporting the formula and planning boundary
QUESTIONS SPECIFIC TO THIS CALCULATION
Questions about projectile unit conversion
Why convert to m/s before resolving components?
A single coherent basis keeps gravity, height, time, and distance compatible and makes the final reconciliation auditable.
Why not use the standard range formula?
That shortcut assumes equal launch and landing elevations. The entered launch height requires the general vertical quadratic.
Does converting speed change the trajectory?
No. Correct conversion represents the same physical speed. Any changed result indicates inconsistent units or premature rounding.
Can I enter a negative launch angle?
Not in this page. It is scoped to launches initially above horizontal; a downward-launch tool needs a different input boundary.
Why are both metres and feet shown?
They support comparison and reporting, while the underlying trajectory remains an SI calculation.
How many decimals should I report?
Match the least precise measured input or the governing reporting standard; retain unrounded values for internal checks.
Why can a horizontal launch have nonzero flight time?
When initial height is above the landing datum, gravity still requires time to bring the projectile down even though initial vertical velocity is zero.
What happens near a 90-degree launch angle?
Horizontal velocity approaches zero, so the ideal range approaches zero while flight time can remain substantial. Display rounding can hide the small component.
When should I avoid this conversion model?
Avoid it when drag, wind, lift, spin, terrain, thrust, or a moving launch platform materially affects the path; use an appropriate numerical or measured model instead.
Why retain velocity components as well as speed and angle?
The components are the quantities used by the equations and provide a direct inverse check when the converted angle or magnitude appears suspicious.
IMPORTANT BOUNDARY
Educational mechanics, not a safety envelope
This calculator is an idealized mechanics and unit-conversion aid. It must not be used to set weapon ranges, public safety zones, structural clearances, or certified sports measurements without an appropriate validated model and qualified review.