Falling paper and metal

Problem

Falling through air. A paper ball and a metal ball of the same size fall from $2\;\text{m}$. Explain in five lines why they reach the ground at different times, despite $g$ being the same for both.

Solution

The acceleration of gravity $g$ really is the same for both: if they fell in a vacuum, without air, they would reach the ground at exactly the same instant, with time:

$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \cdot 2}{9{,}81}} \approx 0{,}64\;\mathrm{s}$

In air, however, air resistance also acts, a force that opposes the motion and grows with velocity. The two balls have the same size (same frontal area), so they experience similar resistance at the same velocity; but the paper ball has a much smaller mass.

For the paper ball, air resistance soon becomes comparable to its weight: it quickly reaches a terminal velocity and its fall slows down. The metal ball, much heavier, “feels” the air very little and keeps accelerating almost as it would in a vacuum.

This is why the metal ball reaches the ground first: not because it has a different $g$, but because air resistance slows the light paper ball down much more.

$\ev{\text{in a vacuum equal times; in air the paper is slowed by resistance}}$

Collegamenti

Argomenti: Kinematics Concetti: Free fall