Trajectory Calculator
Use this trajectory calculator to find the flight path of a projectile. Type in three values — initial velocity, launch angle, and initial height — and instantly find the trajectory formula, its parabolic shape, horizontal range, time of flight, and maximum height. Supports both metric (m/s, m) and imperial/US customary (ft/s, ft) units.
Trajectory Definition
Trajectory, also called a flight path, is the path followed by a moving object under the action of gravity. Usually, we use the term when talking about projectiles or satellites (where the regularly repeating trajectory is called an orbit). If you throw an object for a short distance, then a parabola is a good approximation of the trajectory shape.
Some common examples of projectile motion are:
- Ice hockey puck, baseball, or golf ball in flight.
- A bullet fired from a weapon.
- A jet of water from a fountain.
- An object thrown from a table, plane, building, or bridge.
- A soccer ball kicked at an angle.
- A basketball thrown toward a hoop.
Trajectory Formula
The trajectory formula connects the horizontal position x to the vertical position y directly, without involving time. It is derived from the two equations of motion:
- Horizontal: x = Vx × t
- Vertical: y = h + Vy × t − ½ × g × t²
Since the velocity vector V₀ forms a right triangle with its components:
- Horizontal velocity: Vx = V₀ × cos(α)
- Vertical velocity: Vy = V₀ × sin(α)
Eliminating time by substituting t = x / Vx into the vertical equation gives the trajectory formula:
y = h + x × tan(α) − g × x² / (2 × V₀² × cos²(α))
where:
- y — vertical position (height) at horizontal distance x
- h — initial height above the ground
- x — horizontal distance from launch point
- α — launch angle above horizontal
- V₀ — initial launch velocity
- g — gravitational acceleration (9.81 m/s² or 32.174 ft/s²)
Key Calculated Values
| Quantity | Formula | Description |
|---|---|---|
| Horizontal Range (R) | R = Vx × T | Total horizontal distance traveled when the projectile lands |
| Time of Flight (T) | T = (Vy + √(Vy² + 2gh)) / g | Total time the projectile is in the air |
| Maximum Height (Hmax) | Hmax = h + Vy² / (2g) | Highest point reached during the flight |
| Time to Peak | tpeak = Vy / g | Time to reach the maximum height |
Trajectory Calculator: How to Use
- Choose your unit system — select Metric (m/s, m) or Imperial (ft/s, ft).
- Enter the initial velocity (V₀) — the speed at which the projectile is launched. Must be greater than zero.
- Enter the launch angle (α) — the angle above horizontal at the moment of launch. Must be between 0° and 90°.
- Enter the initial height (h) — the height above the ground at launch. Use 0 for ground-level launch. Cannot be negative.
- Click Calculate — the calculator instantly shows the trajectory formula, range, time of flight, maximum height, velocity components, and a visual parabola chart.
Unit Systems
| Quantity | Metric (SI) | Imperial / US Customary |
|---|---|---|
| Velocity | m/s | ft/s |
| Distance / Height | m | ft |
| Gravity | 9.81 m/s² | 32.174 ft/s² |
FAQs
What is the shape of a trajectory?
Under ideal conditions (no air resistance), the trajectory of a projectile is a parabola. The parabolic shape comes from the constant downward acceleration of gravity combined with constant horizontal velocity. The trajectory formula y = f(x) is a second-degree polynomial in x, which is the equation of a parabola.
What angle gives the maximum range?
When launched from the ground (h = 0), a launch angle of 45° produces the maximum horizontal range. If the launch height h > 0, the optimal angle is slightly less than 45°. The optimal angle can be found by maximizing the range formula with respect to the angle.
What is the difference between trajectory and projectile motion?
Projectile motion refers to the overall physical phenomenon of an object moving under gravity after being launched. The trajectory is specifically the geometric path (the curve) that the projectile traces through space. The trajectory calculator focuses on the path equation y = f(x), while a projectile motion calculator usually focuses on time-based quantities.
Does air resistance affect the trajectory?
Yes — in reality, air resistance (drag) reduces the range and changes the shape of the trajectory from a perfect parabola to a flatter, asymmetric curve. This calculator assumes no air resistance, which is a valid approximation for dense, slow-moving objects over short distances, but less accurate for bullets or long-range artillery.
Can I calculate the trajectory for a projectile launched from a height?
Yes. Simply enter a non-zero value in the Initial Height field. The trajectory formula and all results automatically account for the height offset. Common examples include a ball thrown from a cliff, a cannonball fired from a tower, or a ball rolling off a table.
What is the trajectory at 0° angle?
At 0° launch angle with h = 0, the object is already on the ground and has no vertical velocity — there is no trajectory. However, if h > 0 (launched horizontally from a height), the trajectory is a half-parabola, which is the classic horizontal projectile motion case.