A couple of posts ago, we discussed the idea of Newton’s Third Law: for every action, there is an equal and opposite reaction. Let’s figure out how to mathematically apply that to getting a rocket off the ground and investigating how airplanes fly!

The main idea with thrust is that if you throw enough stuff out the back at a high enough velocity, it will propel you forward. It is therefore likely that the thrust that is experienced is somehow related to the velocity at which the material is expelled and the mass of the expelled material. This is almost exactly right, except that when we are talking about a rocket, or even an airplane, the stuff that is coming out is a stream of material, and so the mass is not really just a mass, but a mass flow rate (i.e., the amount of stuff per second). In other words, thrust is equal to the mass flow rate times the exhaust velocity, or:

For a normal, everyday rocket, the exhaust velocity is roughly equal to about 3,000 m/s (meters per second, or about 6,700 MPH). This exhaust velocity doesn’t vary very much, and it could be as low at 2,000 m/s or as high as about 4,000 m/s, but those are about the limits for a “normal” rocket engine. I will talk about why in further posts, but it mostly has to do with chemistry.

On the other hand, the mass flow rate can vary a large amount from rocket to rocket. That is because the only knob the rocket manufacturers have (if the exhaust velocity is about fixed) is the mass flow rate. A smaller engine will have a smaller mass flow rate, and there won’t be able to lift large things off the ground. A large rocket, like the Saturn 5, the largest successfully flown rocket ever, has a huge mass flow rate, and was able to lift a gigantic amount of stuff off the ground (elephants in space?).

My favorite example of a rocket is the V2, which was the first really “useful” liquid propulsion rocket ever made. This was the rocket build by the Germans in WWII to bomb the Allies. I will discuss these in more detail in another post also. But for now, I would like to use this as an example to look at some aspects of thrust and getting a rocket off the ground. So, let’s get the stats on a V2 rocket:

- With no fuel and no warhead, it had a mass of about 12,500 kg;
- The warhead was about 1,000 kg;
- It carried about 8,800 kg of fuel;
- The mass flow rate about 110 kg/s; and
- The exhaust velocity of the engine was about 2,400 m/s.

With this information, we can calculate some things about the rocket. For example, the most important thing to calculate is whether the rocket can actually lift itself off the ground. To do that we calculate:

- The total mass of the rocket at launch, which is equal to the mass of the rocket, the warhead, and the fuel, which is 22,300 kg.
- The weight of the rocket, which is just the mass times the acceleration due to gravity (-9.8 m/s²), which would then be -218,540 N. It is negative, since gravity is pulling down on it. In order to lift off the ground, the rocket engine has to produce more than 218,540 N of thrust. That will just overcome gravity, and cause the rocket to lift into the air.
- The thrust of the rocket, which is given by the formula above. We know that the exhaust velocity is 2,400 m/s, and the mass flow rate is 110 kg/s. The thrust is therefore 2400*110 = 264,000 N, which is larger than 218,540, so the rocket will definitely move upwards.

In order to actually calculate that rate of acceleration, we can do a bit more math, using Newton’s second law, which is F = m * a, or force equals mass times acceleration, but we can rearrange it to calculate the acceleration (a = F/m). The force is the total of the weight of the rocket (-218,540N) plus the thrust of the rocket (264,000N), which is 45,460N. The total mass of the rocket is 22,300 kg, so the acceleration is 45460/22300 = 2 m/s² upwards.

Now, the cool thing about this has to do with the mass flow rate. The mass flow rate literately means that the rocket is becoming less massive every second. For the V2 rocket, it is becoming 110 kg less massive every second. So, after one second, the rocket has a mass of 22,190 kg. After two seconds, it has a mass of 22,080 kg. And so on, for about 80 seconds, until the rocket has a final mass of 13,500 kg. During the final second of thrust, the acceleration will be quite different than the first second. In the last second, the rocket weighs about -132,300N, while the thrust is exactly the same as before (264,000N), so the difference is 131,700. The mass of the empty rocket is 13,500 kg, so that the acceleration just before the thrust cuts off is about 9.8 m/s² upwards, about 5 times larger than the starting acceleration! This increasing acceleration effect happens with all rockets that have a constant mass flow rate – as they use more fuel up, the accelerate upwards faster and faster. It is no real surprise then that when rockets are sitting on the launch pad and fire their engines, they look like they are just crawling upwards – because they are! As the rocket loses more and more fuel, it gains speed rapidly.

For manned missions, it is important that the acceleration not become too large, since people can’t withstand huge accelerations. People tend to pass out when they are accelerated at rates that are about 5 times Earth’s gravity. And the point of maximum acceleration doesn’t happen at the beginning, but near the end of the thrust.