Here's a video of the Apollo 11 launch, using a Saturn V rocket to boost Neil Armstrong, Buzz Aldrin, and Michael Collins into orbit around the Moon.
The formula for (non-relativistic) momentum is this:
In class, we discussed that our formula for impulse (change in momentum) is typically given as this:
But there is an assumption – this formula assumes that mass doesn’t change. In the case of a rocket, the mass does change. Technically, the formula for impulse should be:
(This is essentially cheap calculus – the only difference is that the capital delta will become a lowercase delta: d).
For this class, we can treat the exhaust velocity v as roughly constant, leaving us with this:
Now, let’s return to our definition of force as the ratio of impulse over a time interval:
Replacing the impulse with what we derived for a rocket,
This is our rocket equation. The force, or thrust, is equal to the rate of mass ejected from the rocket times the velocity of the ejected mass. This velocity is called the exhaust velocity.
The Saturn V rocket was the largest, most powerful rocket ever built by the United States. (It remains the largest and most powerful launch vehicle ever brought to operational status by height, weight, and payload.) It was made with one goal: to send American astronauts to the Moon.
Let’s now derive the acceleration at liftoff of the Saturn V rocket. (From Chapter 9: Linear Momentum and Collisions)
It had a mass of
or 2850 tonnes. Keep in mind that this rocket existed to send about 32 tonnes (the Lunar and the Command/Service modules) to lunar orbit. (Rockets are not really efficient.)
The rate of mass ejected per unit time was
Or 13.8 tonnes. This is about the mass of three elephants. So the Saturn V throws out three elephants worth of mass every second.
The exhaust velocity for a Saturn V was
This is actually faster than a speeding bullet (800-1500 m/s).
So, the thrust generated by the rocket would be
The mass of the Saturn V at launch was
Which means that the weight (gravitational force) would have been
Therefore, the net force on the rocket was
And the acceleration of the rocket at liftoff would have been
If you go back to the video of the Apollo 11 launch, you can see that the initial acceleration does, in fact, appear to be 2 m/s.
This is not much. Right at liftoff, an astronaut would feel a gentle tug – a feeling of being about 20% heavier. However, as the rocket lost mass, the acceleration would become greater, until the astronaut experienced several “gees”, or acceleration several times that due to surface gravity of the Earth. Astronauts therefore require excellent physical fitness to preserve their ability to function during liftoff and landing.
In class, I discussed the history of the modern rocket programs in China, the United States, and Germany. History is packed with examples of how choices and mistakes made by individuals and governments completely change the world.
Qian Xuesen, the father of the Chinese rocket program, was initially trained as an engineer focused on railroad administration. He was able to study mechanical engineering at MIT and learn about applied rocketry thanks to a Boxer Rebellion Indemnity Scholarship, a program created from surplus funds that were paid to the US by the defeated Qing empire. Qian was able to study at Caltech under von Karman and served as a commissioned colonel in the US Army Air Force. He worked with Von Karman to interrogate captured German scientists – Qian actually interrogated Wernher von Braun, who would later be influential in the US rocket program during the Cold War.
In 1949, allegations were made against him that he was a Communist, and he lost his security clearance. He was detained, and eventually placed in what amounted to house arrest. Eventually, he was traded to China for a dozen American fighter pilots captured during the Korean War. He went on to develop China’s rocket program and revamped Chinese education. He lived to see China launch an astronaut with its own rockets and return him safely to Earth.
I do not have time to tell you all the stories I have learned about the history of rocket and space programs in China, Germany, Russia, and the United States. But it is a field filled with people who were attracted by goals inspired, not coincidentally, by Jules Verne and visions of a distant future when humans would not be bound to the Earth of their ancestors. It was, by and large, a peaceful vision, one made more and more imperfect by the necessity to tie rocket development to military applications.
I leave you with a part of John Kennedy's Moon Speech at Rice University. Regardless of the many, many reasons America went to the moon, this serves as a reminder of why so many Americans contributed their genius and grit to such an amazing project, and what we can yet do, if our vision is clear, our minds well-stocked, and our spirits aflame.
