Rocket Science — Basics — Part 2 — Rocket Equation🚀🧑‍🚀

In the previous post, we saw basic terminologies used in rocket science.

If you have not read my part — 1 of Rocket Science basics series, Please go to this link and refresh it once.

Rocket Science — Basics — Part 1🚀🧑‍🚀

In this post, we will dive into Rocket Equation.

Rocket equation is nothing but The maximum speed the rocket can achieve by burning its fuels`. Let’s say you are in a boat filled with large stones as depicted in the below image. If you want move backward, you just need to throw the stones in the forward direction. If you keep throwing the stones into water, you will move at a faster pace because the boat is getting lighter and lighter. It is analogous to rocket also because as the fuels are getting burnt, the rocket will move forward with high pace because it is getting lighter and lighter.

Rocket Equation depiction Courtesy: Wikipedia

Before moving to deriving rocket equation, we need to know some basic terminologies.

Combustion — It is a chemical process in which a substance reacts rapidly with oxygen and gives off heat. The substance is called as fuel and the source for the oxygen is called as oxidizer. During the combustion process, new chemical substances can be formed and that is called as Exhaust. The amount of exhaust can be controlled depending upon the needs. Oxidizers are usually stored in a different tank than the fuel and released in the correct proportion when the rocket is fired.

Propulsion System — It is a machine that generates thrust to make the object move forward. There are three parts to the propulsion system.

1. Rocket engine — It is the main part where the combustion takes place with the fuel and the oxidizer in the combustion chamber.
2. Nozzle — It accelerates the flow of exhaust. Consider, we have a water bottle with cap opened fully and you push with some force. The water will not accelerate as you wanted. Instead, if you put a hole in the cap and push the bottle now, the water comes out of the bottle with so much velocity. This is analogous to Nozzle.
3. Tanks and Pumps — Tanks store the fuels can be liquid or solid. Pumps allows the fuel and oxidizer to be mixed in the combustion chamber.

The propulsion system works by applying Newton’s third law of motion, which states that for every action, there is an equal and opposite reaction. The faster the rocket burns its fuel, the greater the thrust and acceleration.

Exhaust Velocity — The speed at which the exhaust gases comes out of the propulsion system in rockets or in jets. The exhaust gases can be (liquid hydrogen and oxygen — as used by NASA for its liquid chemical rockets or it can be experimented at home by using soda and vinegar).

In the below given image, we can see that the fuel and oxidizer

Space and Soda rocket Courtesy: Yale University

There are two types of exhaust velocity.

1. Actual Exhaust Velocity — The real speed of exhaust gases that comes out of the propulsion engine. For commercial airplanes the actual exhaust velocity would typically be 340 m/s.
2. Effective Exhaust Velocity — Unlike actual exhaust velocity, which measures the real speed of exhaust gases leaving the engine, effective exhaust velocity is more about efficiency. It’s a theoretical number that helps engineers understand how well an engine converts fuel into thrust. Specific Impulse — The amount of thrust generated by the rocket per unit of fuel. This specific impulse is multiplied by acceleration due to gravity (earth’s gravitational force — 9.81 m/sec²).

The exhaust velocity varies depending upon the type of propellant (fuel — either solid or liquid) which we will cover in subsequent posts.

Rocket Equation:

Let Δv be the change in velocity. Let’s say, rocket is ready to be launched from its launch pad (assume it is in Sriharikota and ISRO is launching the rocket). At this point in time, the velocity is v = 0 since it is stationary.

Then, if we assume the rocket is in the lower earth orbit, the rocket will move at nearly 7 km/sec. If the rocket needs to escape the gravitational pull of earth and reach outer space, it should go beyond 11 km/sec. This velocity is called as Escape velocity.

Now, Δv = 7 km/sec if in the low earth orbit and Δv = 11 km/sec while breaching the earth’s gravity.

Now, we can see that in order to test our rocket, whether it will reach speeds like above mentioned, we need the rocket equation. It also explains how much fuel we need to reach those velocities that we have mentioned above.

There are two types of mass one is dry mass (md), fuel mass is (mf) and wet mass (mw). Now, wet mass can be written as mw = md + mf.

We say what is exhaust velocity and we shall represent it as ve.

We can write the rocket equation in two forms:

1. Δv = ve + ln (mw / md) — where ln is the natural logarithm. This helps us to find out the maximum velocity of the rocket.
2. mw = md * exp (Δv / ve) — This equation helps to find out how much fuel we need to change the velocity by a certain amount.

Let’s see rocket equation with an example:

Let say we have wet mass (mw) as 100,000 kg (including fuel), dry mass as 50,000 kg and we have exhaust velocity (ve) for say chemical rocket (we will discuss in depth is subsequent posts) is 3,000 m/s.

Δv = 3000 + ln (100000 / 50000)

Δv = 2,079m/s

Now, we shall be sure that this rocket will not reach the escape velocity to march into outer space. This rocket may be used for ballistic missile which is it will go up and down.

It will also helps us to understand why we need multistage rockets especially to carry satellites to space.

Rockets that carry satellites to earth orbit will be occupied by fuel 96% of its total weight. That means, mw = 96% and md = 4% and by using multistage rockets we will lose weights all the way in order to accelerate and move into earth’s orbit.

Multistage rockets Courtesy: cheez burger

The idea is to use the split the fuel (along with oxidizer) into different tanks. Once the fuel in the last tank is exhausted, that tank will be released. Each tank in a rocket is called as stages. So, we essentially drop a particular stage when fuel is over.

So, we can calculate rocket equation for the first stage and subsequently for the other stages and add up to get the Δv.

By using multi-stage rockets, we will be getting higher Δv rather than with single stage rocket using rocket equation. Let us see with an example.

We have a single stage rocket and the total mass of the rocket is 100kg. The payload of the rocket say a satellite is 10kg. The fuel weight is 80kg and the fuel tank weight is 10kg. The exhaust velocity is 2.5 km/s.

If we calculate the Δv using the rocket equation, we will get 2.5 * ln (100 / 20) [Δv = ve + ln (mw / md)] i.e., the wet mass mw is 100kg and md is 20kg. The output we will get is 4 km/s.

Now, we shall try the same rocket but with two stages. We will split the fuel into 2 tanks with each 40kg. The tank sizes is also halved into 5 kg each. The Δv1 = 2.5 * ln (100 / 60) = 1.3 km/s. Then we drop the first stage and then calculate the Δv2. Now, the Δv2 = 2.5 * ln (55 / 15) = 3.2 km/s. Now, Δv = Δv1 + Δv2 = 3.2 + 1.3 = 4.5 km/s.

So, using one stage Δv is 4 and using two stage, Δv is 4.5

So, we can see that, by using multiple stages in the rocket, we will be able to achieve high Δv that can take rocket to escape velocity.

Animated version of rocket around earth’s orbit

We will discuss about Orbits in next post. Stay tuned.

Thanks for reading!!!