What this calculator does
A trajectory calculator works out the path a launched object follows under gravity alone, once it leaves the ground at a given speed and angle. This covers everything from a thrown ball to an artillery shell to a stream of water from a hose, as long as air resistance is ignored, which is a good approximation for many everyday objects over short distances.
The three headline results, range, maximum height and time of flight, all follow from the same two starting numbers: launch speed and launch angle. Splitting the initial velocity into horizontal and vertical components is the key step, since gravity only acts on the vertical component while the horizontal component stays constant throughout the flight.
The formula
Launch velocity splits into a horizontal component (v·cosθ) that stays constant throughout the flight, and a vertical component (v·sinθ) that gravity steadily slows, stops, and reverses. Range is v²sin(2θ)/g, maximum height is v²sin²θ/2g, and time of flight is 2v·sinθ/g, all assuming launch and landing happen at the same height.
| Term | Meaning |
|---|---|
| Range | The total horizontal distance covered before the object lands back at launch height. |
| Maximum height | The highest point reached during the flight, at the moment vertical velocity hits zero. |
| Time of flight | The total time from launch to landing, assuming a level launch and landing point. |
The inputs explained
| Field | What to enter |
|---|---|
| Launch speed (m/s) | Launch speed, the total speed at the moment of launch, before splitting into horizontal and vertical components. |
| Launch angle (°) | Launch angle, measured from the horizontal ground, where 0° is straight along the ground and 90° is straight up. |
| Gravity (m/s²) | Gravitational acceleration; leave at 9.80665 m/s² for Earth unless modelling a different environment. |
When to use it
Finding the optimal launch angle for range
For a given launch speed, range is greatest at a 45° launch angle; angles above or below 45° in equal steps produce identical, shorter ranges, which is a useful check when maximising distance is the goal.
Clearing an obstacle of known height
Comparing the maximum height reached at a given angle and speed against a known obstacle height shows whether that launch would clear it, before adjusting angle or speed to compensate.
Estimating flight time for timing or safety purposes
Sports, ballistics and safety-margin calculations often need to know how long a projectile stays airborne, which this calculator reports directly as time of flight.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does trajectory change with launch angle, at a fixed speed?
A fixed launch speed, across a range of launch angles.
| Launch angle | Range | Maximum height | Time of flight |
|---|---|---|---|
| 15° | 31.87 m | 2.13 m | 1.32 s |
| 30° | 55.19 m | 7.97 m | 2.55 s |
| 45° | 63.73 m | 15.93 m | 3.61 s |
| 60° | 55.19 m | 23.90 m | 4.42 s |
| 75° | 31.87 m | 29.73 m | 4.92 s |
| 89° | 2.22 m | 31.86 m | 5.10 s |
How does trajectory change with launch speed, at a fixed 45° angle?
A fixed launch angle, across a range of launch speeds.
| Launch speed | Range | Maximum height | Time of flight |
|---|---|---|---|
| 10 m/s | 10.20 m | 2.55 m | 1.44 s |
| 15 m/s | 22.94 m | 5.74 m | 2.16 s |
| 20 m/s | 40.79 m | 10.20 m | 2.88 s |
| 25 m/s | 63.73 m | 15.93 m | 3.61 s |
| 30 m/s | 91.77 m | 22.94 m | 4.33 s |
| 40 m/s | 163.15 m | 40.79 m | 5.77 s |
Questions
What launch angle gives the longest range?
45°, assuming launch and landing occur at the same height and air resistance is ignored. Any angle above or below 45° by the same amount (such as 30° and 60°) produces identical, shorter range, even though the height and flight time differ between them.
Does this account for air resistance?
No, this uses the standard idealised projectile motion model, which assumes gravity is the only force acting once the object is launched. Real-world drag reduces range and height compared with these figures, more noticeably for light or fast-moving objects.
What if launch and landing heights are different?
These formulas assume the object lands at the same height it launched from. A different launch or landing height changes the flight time and range, and needs a version of the calculation that accounts for that height difference separately.
How is this different from the SUVAT equations solver?
The SUVAT equations solver handles general one-dimensional motion problems given any three of five variables. This calculator is specialised for two-dimensional projectile motion specifically, going directly from launch speed and angle to range, height and flight time without needing to set up the horizontal and vertical components separately.
For a general one-dimensional motion problem instead of a full trajectory, see the SUVAT equations solver.