See also parts I, II, III, IV, V, and VI.
Space is sort of hard to get to. You've got one of the Space Shuttle Main Engines (SSMEs), which are really efficient rockets which'll give you a ve of 4.4 km/s in vacuum. That's pretty efficient for burning stuff and about as well as we can do for a rocket that can take off from Earth. But lets say we do want to take off from Earth. Well, plugging that into the rocket equation we see that you need a mass ratio of 8.5 to 1 to get the 9.4 km/s you need to reach orbit. Especially when the big tank you need to hold the hydrogen for your rocket probably has a mass ratio of 10 to 1 before you add in the engines or the shuttle and payload. Thankfully the shuttle also had its boosters which finished burning early then the big casings that contained all that solid fuel were dropped into the ocean where they couldn't slow the rest of the shuttle down.
Being able to drop heavy pieces of your rocket when you're part way to orbit has been a part of rocketry since the beginning. Here's how it works in theory. Lets say you have a rocket that's easy to build and which can carry 1/5 of it's weight as payload. But lets say it only has a delta-v of 5 km/s. Well that doesn't make it to orbit. Ah, but we can make another one that's 5 times bigger and can carry the smaller rocket. We launch the big one, get it up to 5 km/s and it releases the small one which gets its payload up to 10 km/s for a nice high orbit. Our overall mass ratio is 25 to 1. In theory if you just made a single rocket stage with a mass ratio of 25 to 1 that would be just as good - but that's impossible. The tanks you need for fuel limit you to 20 to 1. Add in rocket engines powerful enough to lift the rocket against Earth's gravity and your mass ratio goes down further. You need some sort of staging to get a chemical rocket to Earth orbit.
Early on people couldn't do that sort of theoretical, one rocket carried by another, staging. When you start a rocket on the ground there's gravity pushing the fuel into the engine. When you start a rocket on the ground and the engine just doesn't start you can just fix whatever's wrong and try again. Neither of those is true for upper stages and to start with people didn't know how to deal with that.
What the Soviets and US did was like what the Space Shuttle did. They lit all their engines on the ground and had some high thrust bits that dropped off early while the rest of the rocket made it to orbit. The Soviets with the Sputnik had essentially five identical rocket engines. Four were attached to small tanks and one was attached to a big tank. Under the combined thrust of the five the rocket would be boosted up into the upper atmosphere quickly then four of the engines would burn through their fuel quickly and then just fall away while the last part had a big enough tank in relation to everything else that it eventually produced enough speed to get into orbit.
The US, with the Atlas rocket, did something similar. But instead of having different tanks all three engines were attached to a single fairly efficient tank. The three engines were lit on the ground and loft the rocket up to a high altitude where it would take a long time to come down. Then the two booster engines would fall away and the remaining small and efficient sustainer engine would take its sweet time accelerating the rocket to orbit.
The US and Soviets started figuring out how to make true stages after that. The Soviets put a second stage on top of the rocket with an open cage connecting it to the rest of the rocket. Before the first stage burned out while the acceleration was still forcing the fuel down into the engine they lit it off. The US just used solid rockets where the fuel doesn't need any force to keep it in place and there aren't any turbines to spin up.
Eventually both learned to use further techniques like having little ullage motors produce just enough force to settle the fuel while a new stage was being lighted letting them both use liquid rocket stages that were entirely sequential. Upper stages can use engines that are designed to operate in vacuum with the larger engine bells that let you direct your propellant better but which would cause problems if they had to fight against atmospheric pressure. People talk about taking a single stage to orbit. Elon Musk says that the first stage of his Falcon 9 could just barely make it to orbit if it didn't have to carry other stages or a payload. But there's no reason to go to space unless you're taking something there. Until we develop high efficiency rockets that also produce high thrust we'll have to continue to use staging to make it to space.
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