Free Fall vs Projectile Motion: What Actually Changes

Free fall describes an object moving under gravity alone, along a single vertical line - dropped from rest, or launched straight up or down. Its motion follows the standard constant-acceleration kinematics: v = u + gt for velocity and s = ut + ½gt² for displacement, where g is gravitational acceleration and u is the initial vertical velocity (zero for a simple drop).

A projectile launched at an angle has its initial velocity split into two independent components: a horizontal component (vₓ = v₀ cos θ) and a vertical component (v_y = v₀ sin θ). Ignoring air resistance, the horizontal component never changes throughout the flight - there's no horizontal force acting on it. The vertical component, on the other hand, follows exactly the same free-fall kinematics as a straight vertical drop, just starting from a nonzero initial vertical velocity instead of zero.

That independence is the core idea: horizontal and vertical motion don't affect each other. A projectile fired horizontally from a height and a second object simply dropped from the same height at the same instant hit the ground at the same time - the horizontal velocity of the first one has no effect on how fast it falls.

A projectile launched at 20 m/s at 30° above horizontal has vₓ = 20 cos 30° ≈ 17.3 m/s and v_y = 20 sin 30° = 10 m/s. Using free-fall kinematics on the vertical component alone, with g ≈ 9.81 m/s², time to reach maximum height is t = v_y / g ≈ 1.02 s, and total time of flight back to launch height is roughly twice that, about 2.04 s. Maximum height works out to h = v_y² / (2g) ≈ 5.1 m.

Horizontal range, since vₓ never changes, is just distance = speed × time: R = vₓ × 2.04 ≈ 35.3 m. Every one of those results - time to peak, total flight time, height, range - comes from applying ordinary free-fall kinematics to the vertical component alone, then multiplying the unrelated, unchanging horizontal component by that same time.

For a launch and landing at the same height with no air resistance, range simplifies to R = v₀² sin(2θ) / g. Since sin(2θ) reaches its maximum value of 1 at 2θ = 90°, meaning θ = 45°, that angle produces the greatest horizontal range for a given launch speed - a direct trigonometric consequence of the range formula, not a separate physical principle.

That optimum shifts once launch and landing heights differ. Launching from an elevated point - a shot put, or a projectile fired from a hilltop - favors an angle somewhat below 45 degrees for maximum range, because the extra fall distance below the launch point gives the projectile more time to keep covering horizontal distance, an effect the simple level-ground range formula doesn't capture.

Free fall answers a narrower set of questions - fall time, impact velocity, drop distance - for motion along a single line. Projectile motion adds range (how far horizontally the object travels), maximum height, and time of flight, all of which require combining the independent horizontal and vertical results. If a problem only involves vertical motion (something dropped, or thrown straight up), it's a free-fall problem; if there's a launch angle other than straight up or down, it's a projectile-motion problem that happens to use free-fall physics for its vertical half.