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Drag and speed

Rockets and landing · level 5, lesson 6 of 6

A question to start

In level 3 you saw that drag on a rocket is greatest at "max Q", about a minute after launch. What is Q, and how big does it get?

The idea

The drag force on a body moving through air is

\[F_D = \tfrac12 \rho v^2 C_d A,\]

with \(\rho\) the air's density (kg/m³), \(v\) the speed, \(A\) the area facing the flow, and \(C_d\) the drag coefficient, a number that depends on the shape (level 7 shows where the formula comes from).

The part \(q = \tfrac12 \rho v^2\) is called the dynamic pressure. It has units of pressure (pascals): it is the push of the oncoming air per square metre, before the shape is taken into account. So \(F_D = q\,C_d A\).

For a rocket climbing out of the Earth's air, \(v\) grows fast while \(\rho\) falls fast. Their product has a peak: max Q.

Time after launch (a typical rocket) Height \(\rho\) (kg/m³) Speed \(q\)
30 s 3 km 0.91 150 m/s 10 kPa
60 s 11 km 0.36 400 m/s 29 kPa
90 s 25 km 0.040 900 m/s 16 kPa
150 s 60 km 0.0003 2,000 m/s 0.6 kPa

Max Q is around 30 to 35 kPa for most big rockets: about a third of the pressure of the air around you, pushing on the nose of the rocket. Rockets often throttle down through max Q to keep the stress within the limits of the structure.

Leaving Mars. At Mars's surface \(\rho\) is about 0.020 kg/m³, fifty to sixty times less than at Earth's. A rocket leaving Mars reaches max Q at only about 1 to 2 kPa. Drag hardly matters on the way up from Mars.

Worked examples

1. Max Q. At 11 km and 400 m/s: \(q = 0.5 \times 0.36 \times 400^2 = 28{,}800\) Pa, about 29 kPa.

2. Drag force at max Q. A rocket 3.7 m across (\(A = 10.8\) m²) with \(C_d = 0.5\): \(F_D = 28{,}800 \times 0.5 \times 10.8 = 155{,}000\) N, about 155 kN. Compare with its thrust, several thousand kN: drag is a few per cent.

3. A rocket leaving Mars. At 2 km up, \(\rho \approx 0.017\) kg/m³, at 300 m/s: \(q = 0.5 \times 0.017 \times 90{,}000 = 765\) Pa.

Try it

  1. Find \(q\) for a car at 30 m/s at sea level (\(\rho = 1.225\)).
  2. Find the drag on a car with \(C_d A = 0.7\) m² at that speed.
  3. At what speed would a body in Mars's ground-level air (\(\rho = 0.020\)) feel the same \(q\) as the car?
  4. Why does max Q happen, rather than drag just growing as the rocket gets faster?
  5. Would a rocket leaving Mars need to throttle down for max Q? Why?
Answers
  1. \(0.5 \times 1.225 \times 900 = 551\) Pa.
  2. \(551 \times 0.7 = 386\) N.
  3. \(v = \sqrt{2 \times 551 / 0.020} = 235\) m/s.
  4. As it climbs, the air thins much faster than \(v^2\) grows, so \(q\) rises then falls.
  5. Very unlikely: Mars's air is so thin that \(q\) stays a few per cent of Earth's max Q.

Common mistake

Thinking drag grows in proportion to speed. It grows with the square of the speed: twice as fast, four times the drag (in air of the same density).

Where it leads