Skip to content

Terminal velocity

Air and fluids · level 7, lesson 2 of 6

A question to start

A skydiver on Earth stops speeding up after about 12 seconds and falls at a steady speed. How fast would a skydiver fall on Mars, where gravity is weaker but the air is much thinner?

The idea

A falling object speeds up, so its drag grows. Eventually the drag equals the weight, the forces balance, and it stops speeding up. This top speed is the terminal velocity:

\[\tfrac12 \rho v_t^2 C_D A = mg \quad\Rightarrow\quad v_t = \sqrt{\frac{2mg}{\rho C_D A}}.\]

Weaker gravity lowers \(v_t\); thinner air raises it. On Mars, the thin air wins by far:

\[\frac{v_\text{Mars}}{v_\text{Earth}} = \sqrt{\frac{g_\text{Mars}}{g_\text{Earth}} \times \frac{\rho_\text{Earth}}{\rho_\text{Mars}}} = \sqrt{0.378 \times 76} \approx 5.4.\]
Earth: levels off at 51 m/sMars: heading for 270 m/stimespeed
An 80 kg skydiver, \(C_D A = 0.5\) m² Mars Earth
Gravity 3.71 N/kg 9.81 N/kg
Air density at the ground 0.016 kg/m³ 1.217 kg/m³
Terminal velocity 270 m/s 51 m/s

Worked examples

1. Earth. \(v_t = \sqrt{2 \times 80 \times 9.81/(1.217 \times 0.5)} = \sqrt{2580} = 51\) m/s, about 180 km/h.

2. Mars. \(v_t = \sqrt{2 \times 80 \times 3.71/(0.016 \times 0.5)} = \sqrt{74{,}200} = 272\) m/s, nearly 1,000 km/h, faster than the speed of sound in Mars air.

3. A ping-pong ball (2.7 g, 4 cm across, \(C_D = 0.5\)): about 8 m/s on Earth and 45 m/s on Mars.

Try it

  1. What forces balance at terminal velocity?
  2. How does terminal velocity change if the area is made four times bigger?
  3. A parachute gives an 80 kg skydiver \(C_D A = 30\) m². Find the landing speed on Earth and on Mars.
  4. Why would a person falling on Mars hit the ground harder than on Earth, even though gravity is weaker?
Answers
  1. Weight and drag.
  2. It halves, because \(v_t\) goes with \(1/\sqrt{A}\).
  3. Earth: \(\sqrt{1570/36.5} = 6.6\) m/s. Mars: \(\sqrt{594/0.48} = 35\) m/s.
  4. The air is so thin that drag stays small until the speed is very high.

Common mistake

"Weaker gravity means a gentler fall." On Mars, the thin air means drag hardly slows you, so you reach a far higher speed before the forces balance.

Where it leads