Quick answer
A projectile launched at speed v and angle θ from height h₀ has time of flight t = (v sin θ + √((v sin θ)² + 2gh₀)) ÷ g, range v cos θ × t and peak height h₀ + (v sin θ)² ÷ 2g. At 20 m/s and 45° from the ground: range 40.79 m, flight 2.884 s, peak 10.20 m, impact speed 20 m/s.
Enter the launch speed and angle, and optionally the launch height. The calculator returns range, flight time, maximum height and impact speed. Use this projectile motion calculator to check your own figures.
Related tools: Acceleration Calculator: Average acceleration from change in velocity over time. SUVAT Calculator: Solve constant-acceleration motion for the missing quantities. Free Fall Calculator: Fall time, height and impact speed in free fall from rest.
How to use the Projectile Motion Calculator
- Enter launch speed (m/s) (above 0).
- Enter launch angle (°) (between 0 and 90).
- Optionally, enter launch height (m) (0 or more).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Launch speed | A number, in m/s | > 0 |
| Launch angle | A number, in ° | ≥ 0, ≤ 90 |
| Launch height (optional) | A number, in m | ≥ 0 |
Formula and method
Formula
t = (v sin θ + √((v sin θ)² + 2gh₀)) ÷ g; range = v cos θ × t; peak = h₀ + (v sin θ)² ÷ 2g
IEEE-754 double precision; shown to the decimals indicated.
Ignores air resistance and uses g = 9.80665 m/s².
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Launch speed = 20; Launch angle = 45.
| Result | Value |
|---|---|
| Range | 40.7886 m |
| Time of flight | 2.8842 s |
| Maximum height | 10.1972 m |
| Impact speed | 20 m/s |
Example 2
Inputs: Launch speed = 30; Launch angle = 30; Launch height = 10.
| Result | Value |
|---|---|
| Range | 94.1072 m |
| Time of flight | 3.6222 s |
| Maximum height | 21.4718 m |
| Impact speed | 33.1079 m/s |
Example 3
Inputs: Launch speed = 15; Launch angle = 60; Launch height = 0.
| Result | Value |
|---|---|
| Range | 19.8698 m |
| Time of flight | 2.6493 s |
| Maximum height | 8.6039 m |
| Impact speed | 15 m/s |
The first line is the headline result (range). Below it the projectile Motion Calculator lists time of flight, maximum height and impact speed, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Why 45°
On level ground the range v² sin 2θ ÷ g is greatest at 45°: 400 × 1 ÷ 9.80665 = 40.79 m. Complementary angles (30° and 60°) give the same range.
Idealised
Air resistance is ignored, so real projectiles travel shorter distances. From a height the best angle is below 45°. The impact speed equals the launch speed when launched and landing at the same height.
How the result changes with launch speed
Holding every other input at the first worked example, this table changes only launch speed. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Launch speed | Range | Time of flight | Maximum height |
|---|---|---|---|
| 10 | 10.1972 m | 1.4421 s | 2.5493 m |
| 20 | 40.7886 m | 2.8842 s | 10.1972 m |
| 40 | 163.1546 m | 5.7684 s | 40.7886 m |
| 80 | 652.6184 m | 11.5368 s | 163.1546 m |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Launch speed: entering "0" shows "Launch speed must be greater than 0." beside that field and no result.
- Launch angle: entering "95" shows "Launch angle must be at most 90." beside that field and no result.
Choosing between the Projectile Motion Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Acceleration Calculator](/science/acceleration-calculator/) | initial velocity, final velocity and time | Acceleration |
| [SUVAT Calculator](/science/suvat/) | you know and initial velocity u | Final velocity v |
| [Free Fall Calculator](/science/free-fall-calculator/) | i know the and height or time | Fall time |
| [Magnitude of Acceleration Calculator](/science/magnitude-of-acceleration-calculator/) | x component and y component | Magnitude |
| [Terminal Velocity Calculator](/science/terminal-velocity-calculator/) | mass, frontal area and drag coefficient | Terminal velocity |
Background: motion and mechanics
For constant acceleration the standard relations are v = u + at, s = ut + ½at², v² = u² + 2as and s = ½(u + v)t. Pick the equation that contains your three known quantities and the one you want. Use SI units (metres, seconds, kilograms, newtons, joules) throughout or the answer will be off by a factor.
These equations stop applying when acceleration changes during the motion, as with a rocket burning fuel; that needs calculus or numerical methods.
Assumptions and limitations
- Real projectiles travel shorter distances
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for launch speed and launch angle to confirm this.
Frequently asked questions
What angle gives maximum range?
45° on level ground.
Does mass matter?
Not without air resistance.
What is the range at 30°?
At 30° and 20 m/s the range is v² sin 60° ÷ g = 400 × 0.866 ÷ 9.80665 ≈ 35.3 m, less than the 40.8 m at 45°.
What does the Projectile Motion Calculator calculate?
Range, flight time and peak height of a projectile without air resistance. It returns Range, Time of flight, Maximum height and Impact speed.
What formula does the Projectile Motion Calculator use?
t = (v sin θ + √((v sin θ)² + 2gh₀)) ÷ g; range = v cos θ × t; peak = h₀ + (v sin θ)² ÷ 2g. Ignores air resistance and uses g = 9.80665 m/s².
What do I need to enter in the projectile Motion Calculator?
You enter Launch speed (m/s) and Launch angle (°). Optional: Launch height (m).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Real projectiles travel shorter distances.
What is an example of a projectile Motion result?
With launch speed 20 and launch angle 45, the range is 40.7886 m.
Related calculators
- Acceleration Calculator — Average acceleration from change in velocity over time.
- SUVAT Calculator — Solve constant-acceleration motion for the missing quantities.
- Free Fall Calculator — Fall time, height and impact speed in free fall from rest.
- Magnitude of Acceleration Calculator — Magnitude of an acceleration vector from its components.
- Terminal Velocity Calculator — Terminal speed of a falling object from mass, area and drag coefficient.
- Dew Point Calculator — Dew point from temperature and relative humidity.
Sources
- CODATA 2018 / SI exact constants: g₀ = 9.80665 m/s², c = 299,792,458 m/s, h = 6.62607015×10⁻³⁴ J·s, e = 1.602176634×10⁻¹⁹ C, G = 6.67430×10⁻¹¹ m³/(kg·s²)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.