Problem — Parachute and terminal velocity

Problem

Open parachute. A skydiver in free fall reaches a terminal velocity and then remains at constant velocity. Explain in five lines why, identifying the forces at play.

Solution

Two opposite vertical forces act on the skydiver. Pointing downward is the weight force $mg$, constant and independent of velocity. Pointing upward is air resistance, which grows as velocity increases (for fast bodies in air it is well approximated by a law $F_\text{air} \propto v^2$).

At first, when $v$ is small, resistance is negligible: the weight prevails and the skydiver accelerates. As velocity increases, air resistance grows until it equals the weight.

When $F_\text{air} = mg$, the resultant force is zero, so the acceleration is zero: the velocity no longer changes. This constant value is the terminal velocity. Opening the parachute, by increasing the surface area, suddenly increases air resistance and lowers the terminal velocity to a value safe for landing.

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Topics: Cinematica Concepts: Caduta libera