That’s a thing that has always been bugging me: we need a giant rocket with an ass-load of fuel to get something into the Earth orbit, but when the Eagle launched from the Moon, it merely went poof, and the little fuel the craft could have carried pushed it back into the orbit, with gravity at the moon a sixth of that of the Earth. What is the relation between launch acceleration, amount of fuel and gravity of the respective planet/moon? How much fuel is needed to launch a craft from Mars in a future mission?
Not really an answer to your question: it’s partly a gravity difference, but there is a huge amount of drag just to get out of the Earth’s atmosphere. That alone accounts for a huge amount of fuel. Mars is more of a problem than the moon, as it has some atmosphere and is larger, so more gravity to overcome.
Oh well, I haven’t taken the atmospheres (or lack of any) into account. 
Not sure how accurate this link is, but it gives a rundown of Apollo 11. Orbital velocity for the Earth is about 17,000 mph; for the Moon, 3,600 mph.
That’s still a very fast velocity for a moon launch, and I wonder how the Eagle and later crafts reached that with the little fuel they had.
Calling @Stranger_On_A_Train…
Yeah, he’s also my best hope, but I’m afraid I haven’t seen him around lately…
First of all, it’s not the surface gravitational acceleration that’s relevant; it’s the surface gravitational potential. The surface acceleration is relevant when you’re on the surface, but as you get away from the surface, the acceleration due to gravity decreases. For a big planet, you need to get very far away for a significant decrease, but for a small planet, you don’t need to get very far away. For bodies of the same density (which is approximately true for Terra, Luna, and Mars), the ratio of surface gravitational potentials will be the square of the ratio of surface accelerations: That is to say, if a smaller body has 1/6 the surface gravity, it’ll only have 1/36 the surface potential (the actual ratio for Luna is about 1/22 the Earth’s potential, because the density isn’t exactly the same).
Second, yes, the atmosphere is relevant. For spacecraft, the best atmosphere is none at all, which is what Luna has. That made the Apollo return much easier.
Third, rocket fuel isn’t linear with energy requirements. Launching from Earth means giving the payload 22 times more energy than launching from Luna does, but that means much more than 22 times as much fuel, because you need to launch most of the fuel, too, so you need fuel to launch your fuel (and fuel to launch that fuel, and so on).
Finally, remember that the lunar return vehicle was itself, in its entirety, part of the payload for the Saturn V launch. Of course the Saturn V had to be bigger; it was carrying the other rocket.
Here is an interesting & fun explanation of some of the physics involved.
It doesn’t directly address Moon vs Earth, but it talks about the difference departing Earth between going fast enough to get up to space briefly vs getting to Earth orbit. And thereby covers the exponential increase in fuel to carry the fuel to carry the fuel to carry the fuel to be burned at the end for that last bit of velocity.
Googling around tells me that the total weight of the Saturn V was roughly 580 times the weight of the ascent module of the lunar lander.
Speaking of xkcd, this illustartion is also a good way of showing the differences between leaving Earth, the Moon, and other planets:
Why do we even have science text books?
Just have Randall write them all.
But seriously, if publishers of textbooks are not reaching out to him, for his interesting and easy to understand way of presenting information, they are doing a disservice to our education system.
Similar to what LSLGuy said: Don’t forget that the Saturn 5 weighs an awful lot more than the top half of the Lunar Module!
There’s another good xkcd on orbital speeds:
One of the things Randall explains here is that a lot of the energy expenditure in reaching stable orbit is not in reaching space, but in accelerating laterally to be going fast enough for stable orbit without just falling back to earth. The ISS is moving at about 17,000 mph (the earth’s surface at the equator is moving at about 1,000 mph).
Given this, I find this (amazing) video of a launch seen from the ISS a bit puzzling:
…puzzling because it looks as though it’s just going up. So far as I can find out, this was a resupply mission to the ISS, so it’s heading for the same orbit. All I can think is that it starts by going pretty much straight up, and then it turns to accelerate laterally straight towards us, so we can’t discern that motion?
ETA: on rewatching and seeing how fast the surface is moving away relative to the ISS, I guess it can’t be going straight up even at the beginning. It must be accelerating towards us the whole time.
Thanks! The airglow in that clip is amazing. That’s the yellow band about 100km from the surface, apparently caused by excited sodium.
This. Any movement toward the camera will not be noticeable until the vehicle gets close enough to be significantly changing its apparent size in your field of view.
Picture at top of this article clearly shows curved trajectory. This was commonly seen on smoke trails from space shuttle launches as well, as here. Rockets headed for orbit launch straight up off of the pad, but almost immediately they start leaning over.
I’ve been trying to find out how hard it would be to launch a rocket from Mars, per the OP, and I found this: If It Works, This Will Be the First Rocket Launched From Mars | Air & Space Magazine| Smithsonian Magazine
Not sure of orbital speed, escape velocity, and atmospheric drag for Mars.
That’s really good. Sadly, Estes doesn’t seem to make the E9-4 anymore…
Thanks, that’s a very interesting article. It gives the maximum mass and height of the ascent vessel as 880 pounds and 10 feet, so we have at least ballpark figures to compare it to earth rockets. The whole procedure to get Mars probes to earth and all the little steps involved sound massively complicated and difficult. I hope this will be realized soon.
To put some numbers to it – we can use the unit dV, or Delta Velocity – the change in velocity available to your rocket, IE how much the rocket can increase or decrease its speed by. So if your rocket is moving east at 10 m/s, and it uses 10 m/s of dV to accelerate, it will be moving at 20 m/s.
In low earth orbit, a satellite is moving 7.8km/s. So to go from stationary to orbit, we need to speed up from 0 to 7,800 m/s – but note that we aren’t stationary because of the Earth’s rotation. So going into a polar orbit should take about 7.8 km/s, into an equatorial orbit going east a little less, and going into an equatorial orbit going west a little more. This benefit is greatest at the equator, which is why Florida is a good launch site.
Now, it actually takes almost 10km/s of dV to get into orbit. Where is the extra 2 km/s coming from? Two places.
First, to orbit, you want to move parallel to the surface. But we can’t just burn our rocket straight East, because the atmosphere is in the way. So we burn up. But every second that we are burning up, we are being dragged down by gravity, losing 9.8 m/s of dV every single second. We are wasting dV fighting gravity instead of pushing forwards, towards orbit.
Second, and most significantly, atmospheric drag is fighting us the whole way.
On the moon, with no atmosphere, we have no drag, and we can also turn to burn parallel to the surface as soon as we get high enough to avoid the local terrain features. On Earth, we have to make a slow, gradual turn as we rise through the atmosphere.
On Mars, we will still have to burn up at first, but the atmosphere is much thinner, so drag losses won’t use up nearly as much dV.
Also, on Earth we have to apply that dV slowly as we rise out of the atmosphere; if you took some cargo and accelerated it to 10km/s in a rail gun tube and then shot it at the horizon, it would burn up in the atmosphere.
On the moon, you could build a long magnetic railway, and apply constant acceleration to cargo at justa. Few G’s, then toss the cargo off a cliff at the horizon. Instead of hitting the ground, it would easily get tossed into orbit, and you could power it with local reactors or solar panels to not rely on chemical rocket fuel.