a) The engine thrust is the same (over 7 million pounds).
b) The weight being lifted is the same as a fully loaded Saturn V.
c) We’re not aiming for lunar orbit (or anywhere else)–just speed.
d) Burning the fuel does not decrease the weight.
e) It’s not multi-stage
So, because of “d” and “e”, we’re not constantly getting lighter because we’re constantly burning it up and dropping spent stages. The tanks are still there, but instead of the normal fuels, they’re carrying something else. . . Twinkies, maybe, but the rocket still weighs over 6 million pounds.
There’s a million or so more pounds of thrust than weight, but where would the speed top out? In real life, when the first stage is dropped it’s about 6000 mph.
I’m no mathematician or physicist, but from Wikipedia the Saturn V weighs 3,039,000kg, and each stage provides 34.02, 4.4, and 1 million newtons of thrust for 150, 421, and 500 seconds respectively. So starting in a vacuum, without gravity, and ignoring the mass lost by burning fuel and jettisoning each stage I calculate a total acceleration of 2,364 meters per second, or 5,288 miles per hour. I think that is the maximum velocity if you assume that your magical fuel does not reduce the total mass of the rocket as it burns, and all of the fuel is burned by the first stage.
Unless my math is off by an order of magnitude I think it is clear that weight reduction plays a tremendous role in real world performance, to the point where I am not sure that ignoring it can possibly provide any useful data.
On the one hand you are claiming it’s not a multi-stage rocket, but at the end you state the first stage is dropped at 6,000 mph. You also assume the weight is constant. As the fuel burns, the rocket will get lighter.
It is of course routine to make assumptions in physics, but the trick is knowing when the assumptions are justified. In this case, they’re not. Given that the significant majority of the mass is fuel, and that all of the fuel eventually does get burned, you can’t approximate that the mass is constant and get anything resembling a right answer.
Fortunately, however, all of the complicated calculations have already been made, and we do know the top speed of a Saturn V: It’s the speed at which it actually did go. It was carefully engineered to be able to get to the Moon, which is hard enough as it is, so they had no reason to build additional capability into it. The speed required to reach the Moon, meanwhile, is close enough to escape speed as makes no difference, about 11 km/s.
You’re exactly right, and this is why only multistage vehicles have ever been able to achieve Earth orbit despite decades of effort and billions of dollars trying to do it with a single stage.
I find this topic rather interesting, so I did my best to calculate the actual ideal velocity of a Saturn V. Using the Tsiolkovsky rocket equation and deriving the exhaust velocity from the equation T=(dm/dt)V as described in the wiki article on thrust I calculated the ideal acceleration of each stage individually and added them together to get a final velocity of 8,745 m/s or about 19,560 mph.
Assuming (note: really big assumption this time) I calculated everything correctly that means that the actual acceleration (under ideal conditions) is nearly 4 times that estimated by the simplified equation outlined by the OP.
Now if someone who actually knows what they are doing could check my math that would be stellar. If I am reasonably close I will be super happy.
Oops. On reviewing I just realized that the burn time I cited for the second stage in my first post is wrong, it should be 360 seconds, not 421. Despite the incorrect number cited I double checked the math and my first calculation was correct. I also updated the numbers in this post to reflect the correct burn time.
Thanks all. Anyway in this thought experiment, I was considering only the thrust of the five engines on the first stage, and no weight reduction from burning off fuel.
Hmm, here is the acceleration curve from the Saturn V, but you won’t have much of a curve with constant mass (a jet or rocket is rather more efficient as speed increases, but I’ll leave that out). So figure a constant 12 ms[sup]-2[/sup] acceleration which is all a fully-fuelled Saturn V can manage. Now the second and third stages between them carry about a quarter of the first stage’s fuel load, so that’s an extra 25% duration - let’s call it 200 seconds in total to be generous. Then 200 s x 12 ms[sup]-2[/sup] = 2400 ms[sup]-1[/sup] or pretty close to 5000 mph as a quick and dirty estimate.
Acceleration is speed per time per time. Gravity is 32 feet per second per second. In other words, each second an additional acceleration is imparted and the total velocity increases.
That units you gave are for velocity, which you also mentioned. An acceleration of 5,288 miles per hour would indeed be impressive if kept up for hours. But you never give that part of the equation.
That is the acceleration in the absence of gravity. In a launch from earth, you would have to subtract ~10 ms[sup]-2[/sup], for a net acceleration of ~2 ms[sup]-2[/sup] . As a check, I looked at the actual launch of Apollo 8. Liftoff occurs at 26 seconds; it clears the tower at 36 seconds. The height of the rocket is 110 meters.
So, the actual acceleration of a Saturn V at liftoff is ~2 ms[sup]-2[/sup]. After burning for 200 seconds at this constant acceleration, it would only be going 400 ms[sup]-1[/sup] or 900 mph.
