Projectile Motion Simulator

Kinematics — calculate range, maximum height, and trajectory of a projectile.

Input parameters

v₀

Initial Velocity

Speed at which the projectile is launched

0100

Launch Angle

Angle with respect to horizontal

090

h₀

Initial Height

Height from which the projectile is launched

050

Results

Flight time

—

Maximum range

—

Maximum height

—

Projectile Trajectory

Fundamentals & Explanation

Equations of motion

x(t) = v_0 \cos(\theta)\, t

y(t) = h_0 + v_0 \sin(\theta)\, t - \tfrac{1}{2}g\,t^2

Maximum height

y_{\max} = h_0 + \frac{(v_0 \sin\theta)^2}{2g}

Flight time

h_0 + v_0 \sin(\theta)\,t - \tfrac{1}{2}g\,t^2 = 0

Projectile motion is a special case of two-dimensional motion in which an object is launched with an initial velocity v₀ at an angle θ to the horizontal, under the sole influence of gravity.

It decomposes into two independent motions: uniform rectilinear motion (URM) along the horizontal axis and uniformly accelerated rectilinear motion (UARM) along the vertical axis, with acceleration g = 9.81 m/s² directed downward.

The maximum range from ground level (h₀ = 0) is achieved with a launch angle of 45°, where kinetic energy is equally distributed between horizontal and vertical components.

When the initial height h₀ > 0, the flight time is found by solving the quadratic equation that results from setting y(t) = 0, taking the positive root.

This idealized model neglects air resistance. For real projectiles at high speeds, numerical integration with drag forces is required.

Projectile motion — Wikipedia