Rockets and the rocket equation.

Step 1 of 8

Nothing to push on

In the Scene a rocket stands on its launch pad. Run the motion: it lifts off, burns its propellant, and flies on. Above the atmosphere there is no air and no ground. What does it push off?

Step 2 of 8

Momentum out of the back

In the Unit on momentum we saw that with no outside force, the total momentum is conserved. A rocket and its propellant are one system. When the engine throws hot gas backward at high speed, the gas takes momentum backward, and the rocket gets exactly the same momentum forward, like a skater on ice throwing a heavy ball.

Every second the engine throws out more mass, and the rocket gets lighter. So the same amount of gas speeds it up more as the burn goes on.

Step 3 of 8

The rocket equation

Adding up all those small throws from the start of the burn to its end gives Konstantin Tsiolkovsky’s rocket equation, from 1903:

$\Delta v = v_e \ln\frac{m_0}{m_f}$$\Delta v =$$v_e \ln\frac{m_0}{m_f}$⁠

Here $v_e$⁠ is the exhaust speed, the speed of the gas relative to the rocket; $m_0$⁠ the liftoff mass, with the propellant; $m_f$⁠ the mass at burnout; and $\ln$⁠ the natural logarithm. The ratio $\dfrac{m_0}{m_f}$⁠ is called the mass ratio. $\Delta v$⁠, “delta-v”, is the change in speed the rocket can make in empty space. Gravity and air take some of it on the way.

Show the equation has cut the skin away, and the chart beside the Scene plots $\Delta v$⁠ against Propellant share. All the controls are locked.

Step 4 of 8

The last kilograms

Propellant share $f$⁠ sets the mass ratio: if $f$⁠ of the liftoff mass is propellant, $1 - f$⁠ is left at the end, so $\dfrac{m_0}{m_f} = \dfrac{1}{1 - f}$$\dfrac{m_0}{m_f} =$$\dfrac{1}{1 - f}$⁠. Exhaust speed and Stages are locked.

Step 5 of 8

Exhaust speed

$\Delta v$⁠ is in proportion to $v_e$⁠: an engine whose exhaust is twice as fast gives twice the speed from the same propellant. Exhaust speed depends on the propellant: a solid motor’s exhaust leaves at about 2.6 km/s, kerosene and oxygen at about 3.2, and hydrogen and oxygen, the fastest chemical propellant, at about 4.4 km/s. Propellant share and Stages are locked.

Step 6 of 8

Stages

Until now the tanks and engines rode all the way, even when empty. In a staged rocket each stage is dropped when its propellant runs out, and each stage adds its own $v_e \ln\dfrac{m_0}{m_f}$⁠. Exhaust speed and Propellant share are locked.

Step 7 of 8

To orbit

All the controls are free. There are three ways left to raise $\Delta v$⁠: more propellant, faster exhaust and more stages.

Step 8 of 8

The mass ratio

The question can also be turned round: how much propellant does a given $\Delta v$⁠ take? From the equation, the mass ratio is a power:

$\frac{m_0}{m_f} = e^{\Delta v / v_e}$$\frac{m_0}{m_f} =$$e^{\Delta v / v_e}$⁠

Twice the $\Delta v$⁠ takes the mass ratio squared. This is “the tyranny of the rocket equation”: a little more speed costs a great deal more propellant.