Projectile Motion Calculator

The horizontal range R is the total horizontal distance travelled during the full time of flight. Since horizontal velocity is constant, R = v_x · T_flight. This simplifies to R = v₀² · sin(2θ) / g, which is maximised at θ = 45°.

The impact angle below the horizontal can be found from the ratio of the vertical to horizontal velocity components at landing. By symmetry it equals the launch angle θ.

The horizontal and vertical positions at any time t during the flight are given by the parametric kinematic equations below. These can be used to trace the parabolic trajectory.

Results

The key trajectory results are summarised below. These are the direct outputs of the kinematic equations evaluated for the input launch conditions.

For v₀ = 50 m/s at θ = 40°, the projectile reaches a maximum height of approximately 52.8 m and lands approximately 253 m downrange after about 6.55 s. The impact speed equals the launch speed (≈ 50 m/s) as expected from energy conservation in the absence of air resistance. The impact angle is symmetric with the launch angle at 40° below horizontal.

Note that range is maximised at θ = 45°. At that angle with the same launch speed, the range would be R = v₀²/g ≈ 255 m, slightly greater. The height is maximised at θ = 90° (vertical launch), giving H = v₀²/(2g) ≈ 127.5 m but zero range.

Assumptions & Limitations

• Air resistance (drag) is neglected — results are valid only for dense, slow-moving objects or conceptual/educational use.
• The launch and landing points are at the same elevation (flat ground). For sloped or elevated targets, time of flight and range equations must be modified.
• Gravitational acceleration is assumed constant and uniform — valid for trajectories much smaller than Earth's radius.
• The projectile is treated as a point mass — spin, Magnus effect, and wind are ignored.
• The equations assume a non-rotating reference frame — Coriolis effects are neglected (valid for short ranges).
• Initial conditions assume the projectile is launched from the origin (x = 0, y = 0).
• For very high speeds, relativistic effects are not included (entirely negligible at engineering scales).

Projectile Motion Calculator

This worksheet calculates the full trajectory of a projectile launched at an initial speed and angle above the horizontal, under constant gravitational acceleration and neglecting air resistance.

The governing equations come from Newtonian kinematics. Horizontal and vertical motions are treated as independent: horizontal velocity is constant (no air drag), while vertical motion is uniformly decelerated by gravity. The key quantities derived are time of flight, maximum height, range, and impact velocity.

These equations are fundamental to ballistics, sports science, mechanical engineering (launch systems, catapults), and civil engineering (water jet design, flood trajectory estimates).

The canonical launch equations used here are:

• Horizontal: x(t) = v₀ · cos(θ) · t
• Vertical: y(t) = v₀ · sin(θ) · t − ½ · g · t²
• Time of flight: T = 2 · v₀ · sin(θ) / g
• Maximum range: R = v₀² · sin(2θ) / g
• Maximum height: H = v₀² · sin²(θ) / (2g)

Inputs

Enter the initial launch speed, launch angle above horizontal, and the gravitational acceleration for the environment of interest. Standard Earth gravity is 9.81 m/s². The launch angle should be between 0° and 90° for typical trajectories. An angle of 45° maximises horizontal range on flat ground.

Calculations

First, the launch velocity is resolved into its horizontal (x) and vertical (y) components using basic trigonometry. These components remain independent throughout the flight.

The horizontal component v_x is constant for the entire flight (no horizontal forces act on the projectile). The vertical component v_y is the initial upward speed; gravity continuously reduces it until the apex, then accelerates the projectile downward.

The time of flight T is found by setting y(t) = 0 (projectile returns to launch height) and solving the kinematic equation. This gives T = 2·v_y / g, which is simply twice the time to reach the apex (by symmetry of the parabolic trajectory).

The time to maximum height (apex) is exactly half the total flight time, since the trajectory is symmetric about the apex when launched and landing at the same elevation. At the apex, the vertical velocity is zero.

The maximum height H is the vertical displacement at t = t_apex. Substituting into y(t) = v_y·t − ½·g·t² and simplifying yields H = v_y² / (2g). This is also obtainable from the energy equation v_y² = 2gH.

The impact velocity is the vector magnitude of horizontal and vertical velocity components at landing. By energy conservation (no air drag), the impact speed equals the launch speed. The vertical component at impact is equal and opposite to v_y (downward), and the angle is symmetric.