Trajectory Calculator

Calculate and simulate the trajectory of a projectile based on initial conditions. Key results include maximum height, total distance, flight time, and final velocity.

Trajectory Simulator

Configuration

Initial Height (h)

Launch Angle (θ)

Initial Velocity (V₀)

Key Results

Max Height: -

Total Distance: -

Flight Time: -

Final Velocity: -

Navigating

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Calculator Overview

This trajectory calculator is designed to predict and visualize the flight path of a projectile. It utilizes mathematical models to determine impact points, peak altitude, and duration of flight based on specific starting conditions.

Physics Constraints: The calculator operates within a "gravity-only" environment. While it accounts for practical gravitational physics, it does not factor in external variables such as wind, air resistance, atmospheric pressure, or the aerodynamic shape of the projectile.

Key Inputs and Outputs

The tool allows users to configure the following Initial Conditions:

Initial Height (h): The elevation from which the object is released.
Launch Angle (θ): The angle of release relative to the horizontal ground.
Initial Velocity (v₀): The speed of the projectile at the moment of launch.

The calculator generates the following results:

Maximum Height: The highest vertical point reached during flight.
Total Distance: The total horizontal span traveled (range).
Flight Time: The total elapsed time until impact.
Final Velocity and Angle: The specific speed and angle of the projectile at the point of impact.

Units

Unit Flexibility: To ensure precision across different fields, variables can be entered in multiple units:

Height: Meters or feet.
Angle: Degrees or radians.
Velocity: Meters per second (m/s), feet per second (ft/s), miles per hour (mph), or kilometers per hour (kph).

Notice that the results table will automatically display information in the units selected during your initial configuration.

How It Works

At its core, the calculator solves the fundamental equations of motion. It treats the object’s movement as two independent components:

Horizontal Motion: Calculated at a constant speed, assuming no air resistance.
Vertical Motion: Subject to constant acceleration due to gravity pulling the object back to Earth.

The Standard Trajectory Equation

To map the path, the calculator uses the Equation of Trajectory. This formula is unique because it eliminates "time" as a variable, expressing the relationship between vertical position (y) and horizontal position (x) directly:

y = h + x tan(θ) − gx² / 2v₀²cos²(θ)

Equation Breakdown:

y: Vertical height at any distance x.
h: Initial launch height.
x: Horizontal distance from the launch point.
θ: Launch angle.
g: Acceleration due to gravity (approximately 9.81m/s² or 32.2ft/s²).
v₀: Initial velocity (launch speed).

The Geometry of Flight: Because this is a quadratic equation (y = ax² + bx + c), the resulting path is always a parabola.

The Linear Term (x tan): Represents the path the object would take in a straight diagonal line if gravity were not a factor.
The Quadratic Term (...gx²): Acts as the "gravity penalty." As horizontal distance (x) increases, gravity pulls the object down at an accelerating rate (due to the squared value of x), creating the downward curve.

Related Formulas

While the trajectory equation defines the curve, the calculator utilizes these specific formulas to find key milestones:

Time of Flight: T = 2v₀ sinθ / g (for launches from ground level)
Max Height: H = v₀² sin²θ / 2g
Total Distance (Range): R = v₀² sin(2θ) / g

When a projectile is launched from an initial height (h>0), the flight path is no longer symmetrical. Because the object has further to fall than it had to rise, the standard "ground-to-ground" formula (T = 2v₀ sinθ / g) is insufficient.

Instead, the calculator uses a more comprehensive version of the time of flight formula derived from the kinematic equation for vertical displacement.

The Full Time of Flight Formula

To find the total time (T) until the projectile hits the ground (y=0), the following equation is used:

T = (v₀ sin(θ) + √((v₀ sin(θ))² + 2gh)) / g

This formula is essentially the result of applying the Quadratic Formula to the vertical motion equation 0 = h + (v₀ sin)t − 1/2 gt².

v₀ sin(θ): This is the initial vertical velocity.
√((v₀ sin(θ))² + 2gh): This term accounts for the extra distance and velocity gained as the object falls past its original launch height.
g: The acceleration due to gravity (9.81m/s² or 32.2ft/s²).

Impact on Other Calculations

When the initial height is not zero, several other "standard" assumptions change:

Symmetry is lost: The time it takes to reach the peak is no longer exactly half of the total flight time. It takes longer to fall than it did to rise.
Increased Range: Because the object is in the air longer, it has more time to travel horizontally, resulting in a significantly larger total distance.
Final Velocity: The object will be moving faster at impact than it was at launch because gravity has had more "room" to accelerate it downward.

When a projectile is launched from an initial height (h>0), calculating the Final Velocity and Impact Angle requires breaking the motion into its horizontal and vertical components at the exact moment of impact.

Since there is no horizontal acceleration, the horizontal speed remains constant, while the vertical speed increases as the object falls.

The Components of Final Velocity

To find the total velocity at impact (v_f), you must first determine the final horizontal (v_fx) and vertical (v_fy) components.

Horizontal Component (v_fx): This remains unchanged from the launch.

v_fx = v₀ cos(θ)

Vertical Component (v_fy): This is calculated using the kinematic equation that relates velocity, gravity, and height. Since the object falls a total vertical distance of h (starting from h and ending at 0), we use:

v_fy = √((v₀ sin(θ))² + 2gh)

Notice that on impact, this vector points downward, so it is often expressed as a negative value in coordinate systems.

The Magnitude of Final Velocity

Once you have both components, you use the Pythagorean Theorem to find the total speed (v_f) at the moment it hits the ground:

v_f = √(v_fx² + v_fy²)

Alternative "Energy" Method: If you only need the speed (magnitude) and not the direction, you can use the law of conservation of energy, which is often faster:

v_f = √(v₀² + 2gh)

The Angle of Impact (β)

The angle at which the projectile strikes the ground is determined by the ratio of the vertical speed to the horizontal speed. We use the inverse tangent (arctan) function to find this:

β = tan⁻¹(v_fx / v_fy)

β: The impact angle (measured from the horizontal).
v_fy: The final vertical velocity.
v_fx: The constant horizontal velocity.

Practical Applications

This calculator is an essential tool in any field requiring the prediction of objects (those without independent engines or steering):

Long-Range Shooting & Ballistics: Used by precision shooters to eliminate guesswork. By entering muzzle velocity and gravity constants, shooters calculate the "hold-over" needed to hit targets at significant distances.
Sports Analysis: Coaches use this data to optimize performance. Examples include determining the ideal arc for a basketball free throw to maximize "effective rim size" or finding the release angle for a javelin to achieve maximum distance.
Engineering & Logistics: Used to design industrial collection chutes for conveyor belts or to calculate "re-entry corridors" for aerospace vehicles to ensure they do not bounce off the atmosphere.
Education & Game Development: Students use these tools to work backward from landing points to find initial speeds, while game developers use them to ensure items like grenades or arrows move with realistic physics.