Rocket propulsion

A rocket ascending into space

In recent social media posts, I've noticed a misconception about rocket engines, namely that they propel rockets by pushing off the surrounding air, much like propellers do on airplanes or helicopters. If this were true, rockets would not work in space. But they do work in space. One of the basic maneuvers on missions to the moon is translunar injection (TLI), shown below.

Translunar injection. 1: Rocket is in Earth orbit 2: It accelerates by firing its rocket engine 3: This propels into a highly elliptical orbit that takes it out toward the moon

The rocket initially in is orbit around the Earth (1). TLI is a firing of the rocket engine (2), which accelerates the rocket to much higher speed (2). This propels it into an elliptical orbit that takes it out toward the moon (3).

Rocket propulsion simplified

The rocket engine is a chamber in which solid or liquid fuel is ignited. The combustion produces exhaust gas at very high temperature and pressure. The exhaust gas is allowed to escape through the nozzle at the rear. To see why this propels the rocket, let's think about a simplified model. Think of the rocket engine as a simple cylinder in which the combustion happens. First, imagine a completely closed cylinder that allows no exhaust gas to escape (left in the picture below). This would not create any propulsion because the pressure P would create opposite forces on the ends.

Combustion of fuel in a closed can (left) and in a can that is open on the left end (right)

Now, suppose that the cylinder is open on the left end (right picture), so that the hot gas can escape. Now there's no wall on the left end for the gas pressure to push against. This leads to a force on the right end that is not being cancelled. As gas is expelled to the left, the cylinder is propelled to the right by the unbalanced pressure.

This is a highly oversimplified description of rocket engines. Real rocket engines have a nozzle that tapers down to a bottleneck. This is to better control the flow velocity of the exhaust gas. You can read more on Wikipedia.

Rocket propulsion uses Newton's Third Law

Rocket propulsion uses a principle of physics that works regardless of the specific propulsion mechanism. This is Newton's Third Law, which I introduced in a previous post. The popular statement of the law is that "for every action there is an equal and opposite reaction." The law is poorly understood by many people, partly because this statement is vague and highly confusing. So let's state it more clearly: While object A exerts a force on object B, object B simultaneously exerts a force on object A of equal strength and in the opposite direction. The law works for any transaction between two objects pushing or pulling on each other with some type of force. Examples include the hands of opponents in an arm wrestling match, a cannon recoiling when it fires a cannon ball, a cart recoiling when it shoots a spring-loaded dart, or the collision of an exhaust gas molecule with the wall of a rocket engine chamber.

Bernard de Go Mars, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons

Here's an animation that shows how a boat can be propelled by exploiting the Third Law. The passenger throws rocks from the boat. During each throw, the passenger and the rock are exerting equal and opposite forces on each other, propelling them in opposite directions. Notice that as the passenger uses up his supply of rocks, the boat gets lighter and therefore easier to accelerate.

Now think of each "puff" of hot exhaust gas from the nozzle of a rocket as being like a tossed rock. The expulsion of the gas in one direction, pushes the rocket in the opposite direction. As the rocket uses up its fuel, its mass decreases and it gets progressively easier to accelerate. The key point is that the rocket and the exhaust gas push on each other, not on any other medium in the environment.

Momentum conservation

For a slightly deeper dive, the above examples of cannons, carts, and rockets illustrate conservation of momentum. The momentum of a moving object is the product of its mass and its velocity. A train could be moving slowly but it has a lot of momentum because it's massive. A bullet may have very little mass, but it has a lot of momentum because it's fast.

A force applied to an object for a certain amount time changes its momentum. (More specifically, the momentum change is just force x time. This is an alternative way of stating Newton's Second Law.) BUT momentum, velocity, and force are vectors, which means that they have a direction. For example, if you drive due east at 30 mph for one hour and then drive west at 30 mph for one hour, you end up where you started. The result is no net movement at all. To do the book keeping, we use both positive and negative numbers to describe velocity. We call the eastward velocity +30 mph and westward velocity -30 mph. In this round-trip journey, their contributions to your net movement cancel.

Now let's look at a rocket burn. Let's say that the rocket is at rest before the burn, so it has zero momentum. During the burn the rocket and the exhaust gas exert equal and opposite forces on each other and therefore gain equal and opposite momenta. At the end of the burn, their combined momentum must still be zero, The rightward momentum of the rocket +M v is cancelled by the leftward momentum of the gas -m Ve. The velocity v of the rocket after the burn can be deduced by making this statement mathematically: +M v - m Ve = 0.

A rocket of mass M ejects exhaust gas of mass m. They are propelled in opposite directions with respective velocities +v and -Ve

Using momentum conservation, we can predict how much the velocity of a rocket will be increased by a prolonged rocket burn. The math is simple and requires only two pieces of information: 1. The velocity of the exhaust gas (from the viewpoint of a rocket passenger) once it has completely escaped, and 2. What fraction of the rocket's mass is lost as a result of the burn.

The result is a simple equation called the Tsiolkovsky rocket equation.