PROJECTILE
A parabola, audited — range, apex, and time of flight in closed form.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A projectile leaves the ground at speed v₀ and angle θ. Gravity is the only force acting. How far does it travel, how high does it rise, and how long is it airborne?
Components
Split the launch velocity into components: v_x = v₀cosθ stays constant — nothing pushes horizontally. v_y = v₀sinθ fights gravity, draining at rate g. The two motions are independent.
Solve
Vertical motion sets the clock: the projectile lands when v_y reverses and returns, t = 2v₀sinθ/g. The steady horizontal drift carries it R = v_x·t in that time. Peak height follows from v_y² = 2gh.
Flight
In flight, v_x never changes; v_y falls through zero at the apex and reverses. The velocity vector tilts steadily downward while the horizontal stride stays fixed.
Impact & Symmetry
Impact mirrors launch: the projectile lands at speed v₀, angle −θ — the drop equals the climb. Check the books at the apex: ½v₀² = ½v_x² + gh_max. Mass never appeared; a pebble and a cannonball fly the same arc.