Free fall

Concept & Definition

Drop a stone, and for its entire flight the only force worth counting is gravity. A body moving under gravity alone — no air resistance, no propulsion — is in free fall. “Fall” is slightly misleading: a ball thrown straight up is also in free fall from the instant it leaves your hand, because gravity is still the only influence.

What makes free fall so tractable is that gravity supplies a constant acceleration, directed toward the center of the Earth. Free fall is therefore just constant-acceleration motion wearing a new coat — the same equations, with $g$ in place of $a$.

Units and Dimensions

The acceleration due to gravity near Earth’s surface is $g \approx 9.8\ \text{m/s}^2$, directed downward, with dimensional formula $[\text{L}\,\text{T}^{-2}]$.

Mathematical Formulation

Taking the upward direction as positive (the usual convention), the constant-acceleration equations become:

Equation 1

Velocity in Free Fall

$$v = v_0 - gt$$

where $v$ is the velocity at time $t$, $v_0$ is the initial velocity, and $g \approx 9.8\ \text{m/s}^2$ is the acceleration due to gravity.

Equation 2

Height in Free Fall

$$y = y_0 + v_0 t - \tfrac{1}{2} g t^2$$

where $y$ is the height at time $t$ and $y_0$ is the initial height of the body.

The minus signs are not special to gravity — they simply record that with “up” chosen as positive, the acceleration $-g$ points downward. The body slows as it rises, momentarily stops at the top ($v = 0$), and speeds up on the way down — with the same magnitude of acceleration the whole time.

Common Pitfall

At the highest point of a throw the velocity is instantaneously zero, but the acceleration is not — it remains $-g \approx -9.8\ \text{m/s}^2$. If acceleration vanished at the top, the ball would have no reason to come back down.

Quick Check

A ball is thrown straight up at 19.6 m/s. How long does it take to reach the top, and how high does it go? (Take g = 9.8 m/s².)

Reveal Explanation

Time to the top, using $v = 0$ in Equation 1: $0 = 19.6 - 9.8\,t$, so $t = 2\ \text{s}$.

Height from Equation 2 with $y_0 = 0$: $y = (19.6)(2) - \tfrac{1}{2}(9.8)(2^2) = 39.2 - 19.6 = 19.6\ \text{m}$.