Physics · Mechanics · Grade 10

Jet propulsion and the rocket

What does a rocket push against to fly, and why does it have several stages?

The situation

The rocket that carries an astronaut into orbit has nothing to push against in the void — there is neither air nor ground around it. What, then, does it push off from to accelerate? If you let go of an inflated balloon, it flies forward by throwing the air backward. The water rocket in a school yard, the cart firing balls on rails and the space rocket are all governed by one law. Why does a rocket have several stages, and why do dozens of tonnes of fuel have to be burned to put one tonne of payload into orbit?

Topic

Jet propulsion. From a water rocket to a space rocket: Tsiolkovsky's formula

Key ideas

Conservation of momentum
Tsiolkovsky's formula
Multistage rocket

Formulas

m_{1} {\vec{v}}_{1} + m_{2} {\vec{v}}_{2} = const, Δ v = \frac{m u}{M}

Conservation of momentum: if a piece of mass $m$ is thrown out at speed $u$ , a body of mass $M$ gains a speed $Δ v$ .

m \frac{d v}{d t} = u | \frac{d m}{d t} | + F_{ext}

Meshchersky's equation (a body of variable mass). The reactive force (thrust) is $F_{r} = u | d m / d t |$ .

v = u \ln \frac{m_{0}}{m}

Tsiolkovsky's formula: without external forces, a rocket's speed depends on the exhaust speed $u$ and the mass ratio.

v = u (\ln \frac{m_{0,1}}{m_{k, 1}} + \ln \frac{m_{0,2}}{m_{k, 2}} + \dots)

Multistage rocket: the mass ratios of the stages add up (an empty stage is discarded).

v = u \ln \frac{m_{0}}{m} - g t, v_{1} = \sqrt{g R} \approx 7.9 km/s

Vertical ascent: gravitational loss and the first cosmic velocity.

Theory

Jet propulsion

A rocket throws the gas formed by burning fuel backward at high speed. By the law of conservation of momentum, the rocket gains forward exactly as much momentum as the gas gains backward. So a rocket does not need air — it is “pushed” not by the air but by the matter it expels itself. An inflated balloon, a squid and a jet engine work on the same principle; the recoil (the backward kick) of a rifle when a bullet is fired is also jet propulsion.

A cart firing balls

If a cart (mass with the balls $M_{b}$ ) fires one ball of mass $m$ backward at speed $u$ relative to itself: $M_{b} v = (M_{b} - m) v^{'} + m (v^{'} - u)$ , so $Δ v = m u / M_{b}$ . If we fire all of them at once, $Δ v = N m u / (M + N m)$ . Firing them one at a time gives more: the later balls push a lighter cart. If the balls become smaller and their number grows without limit, the sum tends to $u \ln ((M + N m) / M)$ — this is Tsiolkovsky's formula. The total momentum of the cart and the balls always stays zero.

Meshchersky's equation

In time $d t$ the rocket expels gas of mass $| d m |$ at speed $u$ . From conservation of momentum, $m d v = u | d m | + F_{ext} d t$ , that is $m d v / d t = u | d m / d t | + F_{ext}$ . The first term is the reactive force (thrust): if 1400 kg of fuel per second is expelled at 3 km/s, $F_{r} \approx 4.2$ MN. As the mass decreases, the same thrust accelerates the rocket more and more — the $v (t)$ graph curves upward.

Tsiolkovsky's formula

With no external forces, $v = u \ln (m_{0} / m)$ : the speed depends only on the exhaust speed $u$ and the mass ratio, not on how fast the fuel burns. The logarithm grows slowly: to reach 7.9 km/s with $u = 3$ km/s you need $m_{0} / m = e^{7.9 / 3} \approx 14$ , that is, ≈ 93 % of the rocket must be fuel. But the tanks and engines themselves make up 5–10 % of a stage's mass — and there is a payload too. So a single-stage rocket cannot reach the first cosmic velocity.

Multistage rocket

A stage whose fuel is used up (empty tanks, engines) is discarded, and the next stage works like a new, lighter rocket and again gets a large mass ratio. The speeds add up. Tsiolkovsky called this idea “rocket trains”. On leaving the Earth, gravity “eats” $g \cdot 1$ s ≈ 9.8 m/s of speed every second, and the air resists as well; so rockets soon turn towards the horizontal, and to reach low orbit a total Δv of ≈ 9.3–9.5 km/s is spent. In the model: $\dot{m}$ is constant in each stage, $g$ is constant on leaving the Earth, and there is no air resistance.

Water rocket

Compressed air in the bottle expels the water through the neck at a speed $v_{e} = \sqrt{2 Δ p / ρ}$ (≈ 28 m/s at 4 bar), and the reactive force $F = ρ S v_{e}^{2} = 2 S Δ p$ reaches several hundred newtons. As the air expands, the pressure falls. If there is very little water, the expelled mass is small; if too much, there is little air and the rocket is heavy. The rocket rises highest when about a third of it is filled with water. When the water has gone, the remaining compressed air gives a short extra push. Model: the pressure of the water column and the mass of the air are not taken into account, the air expands adiabatically, $C = 0.4$ .

Did you know?

Konstantin Tsiolkovsky published the rocket speed formula in 1903, in his article “The Exploration of Cosmic Space by Means of Reaction Devices”.
On 12 April 1961 Yuri Gagarin flew in the spacecraft Vostok from the Baikonur Cosmodrome and orbited the Earth once — the flight lasted about 108 minutes.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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