Projectile Motion Calculator

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Range, maximum height and flight time from launch speed and angle.

Inputs

Launch speed (m/s)

Launch angle (°)

Gravity (m/s²)

How to use this calculator: Projectile Motion Calculator

Range, maximum height and flight time from launch speed and angle. The example below is calculated by this page's real engine from the displayed inputs.

1Confirm that the Projectile Motion Calculator matches the quantity or design check you need.
2Enter Launch speed, Launch angle, and Gravity using the units printed beside each field.
3Check every value before calculating, especially decimal points and measurement units.
4Calculate, then follow the substituted equations in the worked example and compare the result with any stated limit.
5Read the assumptions, warnings and cited references before using the result for a financial, medical or engineering decision.

Input guide and example values

Use values from the same measurement basis and time period. Conditional fields appear only when the related option is selected.

Input	Example value	Why it matters
Launch speed	20 m/s	Measured or known launch speed used by the calculation engine.
Launch angle	45 °	Measured or known launch angle used by the calculation engine.
Gravity	9.81 m/s²	Measured or known gravity used by the calculation engine.

Formula inputs & variables for Projectile Motion Calculator

These are the named quantities used by this calculator. When the source formula does not define a mathematical symbol, OneCalcApp keeps the real input label instead of inventing one.

Variable / input	Unit	Meaning in this calculation
Launch speed	m/s	Measured or known launch speed used by the calculation engine.
Launch angle	°	Measured or known launch angle used by the calculation engine.
Gravity	m/s²	Measured or known gravity used by the calculation engine.

How the Projectile Motion Calculator works

The Projectile Motion Calculator uses Launch speed, Launch angle, and Gravity to calculate Range, Maximum height, and Time of flight. Its engine applies R = u²·sin(2θ)/g; the worked values below come from that same live calculation rather than a separately typed example.

With Launch speed 20 m/s, Launch angle 45 °, and Gravity 9.81 m/s², the main worked-example result is Range = 40.77 m.

How each Projectile Motion Calculator input is used

Launch speed

The Projectile Motion Calculator worked example uses Launch speed = 20 m/s. This value is passed directly into the calculation, with an allowed minimum 0.

Launch angle

The Projectile Motion Calculator worked example uses Launch angle = 45 °. This value is passed directly into the calculation, with an allowed minimum 0 and maximum 90.

Gravity

The Projectile Motion Calculator worked example uses Gravity = 9.81 m/s². This value is passed directly into the calculation, with an allowed minimum 0.

Projectile Motion Calculator formulas and result interpretation

Formula 1: relationship used

In the Projectile Motion Calculator, R = u²·sin(2θ)/g. The quantities in this relationship come from the named inputs or from an earlier calculation step shown in the worked example.

Formula 2: relationship used

In the Projectile Motion Calculator, H = u²·sin²(θ)/(2g). The quantities in this relationship come from the named inputs or from an earlier calculation step shown in the worked example.

Formula 3: relationship used

In the Projectile Motion Calculator, T = 2u·sin(θ)/g. The quantities in this relationship come from the named inputs or from an earlier calculation step shown in the worked example.

Range

For the displayed Projectile Motion Calculator worked example, Range is 40.77 m. Verify Launch speed, Launch angle, and Gravity and their units before relying on this output.

Maximum height

For the displayed Projectile Motion Calculator worked example, Maximum height is 10.19 m. Verify Launch speed, Launch angle, and Gravity and their units before relying on this output.

Time of flight

For the displayed Projectile Motion Calculator worked example, Time of flight is 2.88 s. Verify Launch speed, Launch angle, and Gravity and their units before relying on this output.

Projectile Motion Calculator accuracy, checks and limitations

Projectile Motion Calculator units check: confirm Launch speed (m/s), Launch angle (°), and Gravity (m/s²) before calculating.
Projectile Motion Calculator result check: compare Range, Maximum height, and Time of flight with the substituted formula steps and the displayed rounding precision.
Projectile Motion Calculator: Treat the result as a mathematical estimate and separately verify input units, rounding rules and any conventions required for your use case.

Formula, derivation and worked example

Assumes level ground and no air resistance, the standard classroom projectile model.

R = u²·sin(2θ)/g

H = u²·sin²(θ)/(2g)

T = 2u·sin(θ)/g

Substitution steps

1. Range
20²·sin(2θ)/9.81
= 40.77 m
2. Height
20²·sin²(θ)/(2×9.81)
= 10.19 m

Computed example results

Range: 40.77 m
Maximum height: 10.19 m
Time of flight: 2.88 s

Understanding the result

Read the main result together with supporting checks, assumptions, limits and intermediate values.

For a manual check, repeat the first equation, confirm the units and change one input at a time.

Assumptions and calculation scope

Launch speed is assumed to be entered in m/s; converting from another unit before entry avoids changing the numerical meaning of the calculation.
Projectile Motion Calculator applies the displayed R = u²·sin(2θ)/g relationship to the entered math inputs. Factors that are not exposed as inputs or stated assumptions are outside this calculator's calculation scope.

Common mistakes when using Projectile Motion Calculator

Do not mix units for Launch speed (m/s), Launch angle (°), Gravity (m/s²). A unit mismatch changes the input magnitude even when the typed number looks reasonable.
Do not replace the displayed R = u²·sin(2θ)/g relationship with a different convention without also changing the underlying assumptions; compare like-for-like methods when checking the result.
Do not treat Range = 40.77 m from the worked example as a universal answer. It belongs to the displayed example inputs and must be recalculated for the actual case.

When the Projectile Motion Calculator is useful

Projectile Motion Calculator is designed for cases where Launch speed, Launch angle, Gravity are known and you need Range, Maximum height, Time of flight. The page keeps the live calculator, calculation method and worked example together so the result can be checked instead of treated as a black-box number.

Use the calculator for the scope described by its inputs and notes. The displayed method is R = u²·sin(2θ)/g. If the real project or decision needs factors that are not represented here, treat the result as an estimate and add the missing checks separately.

Launch speed and Launch angle: what changes the answer

The worked example uses Launch speed = 20 m/s, Launch angle = 45 °, Gravity = 9.81 m/s². With those values, Range is 40.77 m. Changing an input should be interpreted according to that field's unit, range, option and hint rather than by the number alone.

For this calculator, the main input roles are: Launch speed (m/s): Measured or known launch speed used by the calculation engine. Launch angle (°): Measured or known launch angle used by the calculation engine. Gravity (m/s²): Measured or known gravity used by the calculation engine.

How to sanity-check a Projectile Motion Calculator result

Start by confirming the entered values and units, then compare the substituted working with the displayed formula or calculation steps. Pay particular attention to Range, because it is the first worked-example output shown by the live engine.

Finally, compare the result with the assumptions, warnings and related calculators on this page. A nearby calculator can be useful as a cross-check when it measures the same workflow from a different input or output direction.

Next logical calculator

Continue with Kinetic Energy Calculator

Kinetic Energy Calculator is directly connected from Projectile Motion Calculator as a source-defined continuation or comparison.

Open Kinetic Energy Calculator

Formula

R = u²·sin(2θ)/g
H = u²·sin²(θ)/(2g)
T = 2u·sin(θ)/g

Assumes level ground and no air resistance, the standard classroom projectile model.

Frequently asked questions

What inputs does the Projectile Motion Calculator use?

It uses Launch speed, Launch angle, and Gravity. Follow the unit printed for each field and choose any selectable option to match the real scenario.

What does the Projectile Motion Calculator calculate?

It calculates Range, Maximum height, and Time of flight. With the displayed default inputs, Range is 40.77 m.

Which formula does the Projectile Motion Calculator use?

The primary relationship is R = u²·sin(2θ)/g. The page also shows substituted values and calculation steps so the result can be checked independently.

How can I verify a Projectile Motion Calculator result?

First verify the units for Launch speed, Launch angle, and Gravity. Then compare the substituted formula steps with Range and its displayed precision.

What are the limitations of the Projectile Motion Calculator?

Projectile Motion Calculator: Treat the result as a mathematical estimate and separately verify input units, rounding rules and any conventions required for your use case.

Where can I find related Math tools?

Use the related-tools section on this page to compare another method, change units or continue the same math calculation.

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