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Speeding up

Rockets and landing · level 3, lesson 4 of 6

A question to start

At lift-off a big rocket rises slowly, as if it were struggling. Eight minutes later it is going 7,800 m every second. How does it speed up so much, and why does it get faster more and more quickly?

The idea

How quickly something gains speed is its acceleration, measured in metres per second every second (m/s²). It depends on two things:

  • the net force on it (the leftover push);
  • its mass (how much stuff there is to move).
\[\text{acceleration} = \frac{\text{net force}}{\text{mass}}\]

A bigger push speeds it up faster; a bigger mass speeds up more slowly.

At lift-off, the Saturn V's net force was about 5,000 kN on a mass of about 2,970 tonnes (2,970,000 kg). Acceleration \(= 5{,}000{,}000 \div 2{,}970{,}000 \approx 1.7\) m/s²: slow. After one second it was going only 1.7 m/s, a gentle walk.

As it climbs, it burns propellant at about 13 tonnes every second. Its mass drops fast, but the engines push as hard as ever (in fact a little harder, as the air thins). So the acceleration grows. Near the end of the first stage's burn, the Saturn V was speeding up at about 4 g, nearly 40 m/s every second, and the astronauts were pressed into their seats by four times their weight.

Saturn V, first stage At lift-off Near the end of the burn
Mass about 2,970 t about 800 t
Engines firing 5 4
Thrust about 34,000 kN about 31,000 kN
Acceleration about 1.7 m/s² about 38 m/s² (about 4 g)

(The centre engine was shut down early, so the acceleration would not climb past about 4 g.)

Worked examples

1. A light model rocket. Mass 0.1 kg, net force 5 N. Acceleration \(= 5 \div 0.1 = 50\) m/s²: after 1 second it is going 50 m/s. Small rockets leap; big ones rise.

2. Half the mass. If a rocket burns half its mass and the net force stays the same, its acceleration doubles.

Try it

  1. A rocket has a net force of 2,000 kN and a mass of 1,000,000 kg. Find its acceleration.
  2. If the same rocket's mass falls to 500,000 kg with the same net force, what is its acceleration?
  3. Why does a big rocket speed up faster and faster during its burn?
  4. A rocket has an acceleration of 20 m/s². How fast is it going after 5 seconds, starting from rest?
Answers
  1. \(2{,}000{,}000 \div 1{,}000{,}000 = 2\) m/s².
  2. 4 m/s².
  3. It burns propellant and gets lighter, but the thrust stays the same (or grows), so each second it speeds up more.
  4. \(20 \times 5 = 100\) m/s.

Common mistake

Using the thrust instead of the net force. At lift-off most of the thrust only holds the rocket up against its weight. Only the leftover force speeds it up.

Where it leads