Rockets and Propulsion: Momentum in Action

Pushing on nothing

In 1920, a newspaper editorial famously mocked the rocket pioneer Robert Goddard, claiming that a rocket could not work in the vacuum of space because there would be nothing for it to push against. The editors were wrong, and conservation of momentum shows why. A rocket does not push on the air or the ground. It pushes on its own exhaust.

This lesson applies momentum to one of the technologies your curriculum names: rocket propulsion. You will calculate thrust, see why rockets are mostly fuel and why they are built in stages, and consider ways to improve rockets and the effects of launches on society and the environment.

How a rocket works

A rocket engine burns fuel and blasts the hot exhaust gas out of its nozzle at very high speed. Think of the rocket and its fuel as one system, initially at rest in space:

• The total momentum starts at zero.
• The engine pushes each bit of gas backward. By Newton's third law, the gas pushes the rocket forward.
• These are internal forces, so the total momentum stays zero: the backward momentum of the exhaust is balanced by the forward momentum of the rocket.

A rocket is really a continuous explosion: like the carts pushed apart by a spring in the lesson on explosions and recoil, but happening thousands of times a second.

Thrust

Apply the impulse–momentum theorem to the exhaust. In a time Δt, the engine gives a mass Δm of gas a backward velocity v_ex (the exhaust speed, relative to the rocket). The force needed is F = Δp ÷ Δt = (Δm · v_ex) ÷ Δt. By the third law, the gas pushes the rocket forward with the same force, called the thrust:

F_thrust = (Δm ÷ Δt) × v_ex

So thrust depends on how much mass is thrown out each second and how fast it is thrown. This is Newton's second law in its momentum form, F = Δp ÷ Δt, the form that works when mass changes.

At launch, the rocket rises only if the thrust is greater than its weight: F_net = F_thrust − mg.

Why rockets are mostly fuel

As the rocket burns fuel, it gets lighter, so each kilogram of exhaust changes its velocity a bit more. Adding up all these small changes needs calculus (MCV4U and first-year university). The result is the rocket equation, published by the Russian scientist Konstantin Tsiolkovsky in 1903:

Δv = v_ex ln(m₀ ÷ m_f)

where m₀ is the starting mass (with fuel), m_f is the final mass (empty), and ln is the natural logarithm. You do not need to derive it, but notice what it says:

• Δv grows only with the logarithm of the mass ratio. Doubling the fuel does not double the speed.
• To reach low Earth orbit (about 7.8 km/s, plus extra to overcome air resistance and gravity on the way up) with v_ex ≈ 3 km/s, the mass ratio must be well over 10: a rocket that is more than 90% fuel.
• This is why rockets use stages: when a stage's tanks are empty, the whole stage is dropped, so the remaining fuel does not have to keep accelerating dead weight.

Other momentum engines

• Jet engines take in air, compress it, burn fuel in it, and blast it out faster than it came in. Thrust ≈ (mass of air per second) × (exhaust speed − incoming speed).
• Propellers and boat paddles push air or water backward; the air or water pushes the plane or canoe forward.
• Squid and octopuses squirt water out through a funnel to dart away.
• Ion thrusters on some space probes accelerate charged atoms to very high exhaust speeds (tens of kilometres per second). Their thrust is tiny, but they use very little propellant and can run for months.

Improving rocket technology (C1.1)

Engineers improve rockets by raising v_ex (better fuels, such as liquid hydrogen with liquid oxygen, and better nozzles), reducing the empty mass (lighter tanks and composite materials), using staging, and reusing boosters: some companies now land first stages and fly them again, which cuts the cost and the waste of each launch.

Impacts on society and the environment (C1.2)

Rockets put satellites in orbit that people rely on every day: weather forecasting, communications, navigation (GPS) and Earth observation. Canada's RADARSAT Constellation Mission, for example, uses radar satellites to monitor sea ice, ships and floods. But launches also have costs: rocket exhaust releases carbon dioxide, water vapour and, for some fuels, soot directly into the upper atmosphere; spent stages and debris are left in orbit or fall into the ocean; and launch sites affect local wildlife and communities. Evaluating a launch program means weighing these benefits and costs.