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Projectile motion calculator: range, height and flight time

Find range, flight time, maximum height and impact speed with a trajectory plot under the no-drag model.

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Calculate a projectile trajectory from initial speed, launch angle, initial height and gravity.

Results

Range
-
Flight time
-
Maximum height
-
Impact speed
-

Trajectory

Calculation details

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    How does this tool work?

    Projectile motion combines uniform horizontal motion with vertical motion accelerated by gravity. The calculator resolves the initial velocity into components, finds the time to reach the ground and plots the resulting path.

    Horizontal speed is v₀ cos θ and remains constant when drag is neglected.

    Vertical speed starts at v₀ sin θ and changes under gravity. Flight time is the positive root of y(t)=0.

    A non-zero starting height is supported, so launches from platforms or elevated points are included.

    How to use this tool

    1. Define the launch

      Enter initial speed, angle measured from the horizontal and starting height. Standard Earth gravity is provided but can be changed.

    2. Resolve the velocity

      The engine calculates v₀ cos θ horizontally and v₀ sin θ vertically. In the no-drag model, the two components can be treated independently.

    3. Find flight time

      Vertical position follows y(t)=h+v₀ sin(θ)t−½gt². The positive time root for reaching ground level gives the flight time.

    4. Derive range, maximum height and impact speed

      Horizontal motion gives range, while vertical kinematics gives peak height and the velocity component at impact.

    Good to know

    Horizontal and vertical motion are independent in the model

    With no air resistance, horizontal acceleration is zero while vertical acceleration is −g. That is why horizontal speed stays constant while vertical speed changes throughout the flight.

    45° is optimal only under specific assumptions

    A 45-degree launch maximizes range only when launch and landing heights are equal, gravity is constant and drag is ignored. Elevated launches or aerodynamic effects shift the optimum.

    The plotted path is not a full ballistic prediction

    The graph represents a point projectile acted on only by gravity. It omits drag, lift, wind, spin, object shape and changes in gravitational acceleration.

    Formula

    Projectile motion is split into two independent directions: constant horizontal velocity and uniformly accelerated vertical motion. Combining them reconstructs the trajectory.

    Trajectory equations

    x = v₀ cos(θ)t ; y = h + v₀ sin(θ)t − ½gt²

    Horizontal and vertical motions are solved independently.

    v₀
    initial speed
    θ
    launch angle
    g
    gravitational acceleration

    Assumes constant gravity, level ground at y=0 and no air resistance.

    Use cases

    Mechanics homework

    Check a projectile problem by keeping horizontal and vertical motion separate. The plotted path helps connect the equations with the physical motion and verify range, flight time, peak height and impact speed.

    Sports trajectories

    Explore how launch angle or initial speed changes an ideal ball trajectory. The result is useful for intuition but does not reproduce drag, wind, lift or spin effects.

    Launches from an elevated point

    Model an object projected from a platform, cliff or balcony by using a non-zero starting height. Flight time and range then differ from the same launch at ground level.

    Comparing different gravity values

    Change g to explore the same ideal launch on another celestial body or in a textbook problem that specifies a different constant gravitational acceleration.

    Examples

    45-degree launch

    Using g = 9.80665 m/s².

    Input

    v₀ = 20 m/s; θ = 45°; h = 0 m

    Result

    Range ≈ 40.79 m; time ≈ 2.88 s

    Launch from 10 m

    The path starts above ground level.

    Input

    v₀ = 15 m/s; θ = 30°; h = 10 m

    Result

    Longer flight time

    Tips and best practices

    • The model neglects air resistance.
    • The launch angle is measured from the horizontal.
    • Drag can materially change the range of light or fast objects.
    • When comparing launch angles, keep initial speed and height fixed so that angle is the only changing factor.
    • If the calculated range looks unrealistically large for a light object, air resistance is probably not negligible.

    Sources and references

    Frequently asked questions

    Which angle gives maximum range?

    From level ground with no drag, 45° maximizes range for a fixed initial speed.

    Is initial height included?

    Yes. Time of flight and range both account for the starting height.

    Can I change gravity?

    Yes, which is useful for other planets or textbook exercises.

    Why can a real trajectory differ?

    Air resistance, wind and spin are omitted from this introductory model.

    Why is the trajectory a parabola?

    Because horizontal position is linear in time while vertical position includes a t² term from constant gravitational acceleration.

    Can this calculator include wind or drag?

    No. It uses the standard no-drag model; an aerodynamic model would require additional parameters.

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