Saturn V

Factual Questions
1

I was thinking about how they plan to go to mars someday.
The biggest restriction is time…it’ll take months to years to get there and back.
So basically they need to start their initial thrust to the red planet already in space with a full tank of gas… not a ground launch, but a ship built and fueled in orbit.

Give me a good idea how this can be accomplished.

Lets say they have a fully fueled Saturn V in orbit:
No gravity or atmosphere to work against.

Lets also adjust the rocket nozzle to be optimized for thrust in a vaccuum.

About how fast and how far would the command module be going
after the initial firing of the main engines
(including firing and jettisoning the intermediate stages)?

2

I was going to say “seven miles per second”, because that’s the final velocity of the CSM/LM/dry third stage after lunar trajectory is attained.
But then I realized that the first and second stages of the Saturn have to overcome atmospheric drag and lift the upper stages over a hundred vertical miles, plus the fact that until orbit is reached you’re constantly fighting just to maintain altitude. (If you didn’t accelerate fast enough, you could waste the entire fuel load just hovering against gravity).
So I presume that the total Delta-V is well over seven miles per second, but I have no idea how much more. Tech specs anyone?

3

Remember that a rocket in orbit is not immune to the effects of gravity. It still must overcome the pull of gravity to leave orbit.

4

ALright, lets put it this way.

we put this Saturn V in a perfect void…no air or gravitational fields. Straight line acceleration, use up all fuel in each stage and jettison the stage, lightening the load. Final stage…dock with Lunar Lander, which fires its engine. Jettison Lunar lander and fire engine on Service Module, until out of gas.

After all the engines, stages, and fuel are gone, how far and how fast has the capsule gone, relative to the point of origin where the first stage was fired?

5

It’s been too long since I took the class to remember exactly how this works. I forget the rocket equation and how to calculate it.

Here is the information you need:
http://www.apollosaturn.com/asnr/p9-13.htm

It lists the first 3 stages of the Saturn V, their weights and fuel weights, plus the weight for the Instrument Unit - the total gross weight including payload is at the top. It also has thrust and burn time information for each stage.

The trick is that to compute the acceleration, you have to consider the mass as a changing quantity. First and foremost, you treat each stage independently. Stage 1 has an active part and a payload, which includes the other stages. Etc. Then to really do it right, you have to consider that as you burn fuel, the mass decreases. There’s a nifty formula I learned in one of my fluids classes (I think it was Gas Dynamics), but now I don’t remember any of it.

Have fun with it. :wink:

6

Okay, I played with this some more. Here’s what I came up with. Using the numbers from the previously mentioned web site,

Total Weight WT = 6,200,000 lb

Stage 1:
W1 = 4,792,000 lb
weight empty stage 1 We1 = 4,492,000 lb
burn time tb1 = 150 s
Thrust T1 = 7,500,000 lb

Stage 2:
W2 = 1,037,000 lb We2 = 942,000 lb
tb2 = 359 s
T2 = 1,125,000 lb

Stage 3:
W3 = 262,000 lb We3 = 228,000 lb
tb3 = 480 s
T3 = 225,000 lb

You want to know the final velocity. We know the thrust for each stage, and because we don’t have gravity, we know the sum of the forces is the thrust.

We can also figure the force by Newton’s second law. But remember, the common form assumes constant mass, which we do not have.

F = d(mv)/dt Force equals time derivative of momentum.

Using constant fuel burn rate, this simplifies to

F = Delta (mv) / Delta t, or (m2 v2 - m1 v1)/(t2 - t1)

where m2 is mass at end of burn
m1 is mass at beginning of burn
v2 is velocity at end of burn
v1 is velocity at beginning of burn
t2 - t1 = tb burn time

This must be calculated for each stage, as the masses vary by burning fuel and dropping stages. I’ll let you plug numbers yourself, but I came up with a final velocity

v = 267,967 ft/sec about 183,000 mi/hr

That is first three stages, not including LEM descent and ascent engines. You can get that info from the previous link.

7

FTR, I’m in High School, haven’t taken Calculus or even Physics 30. Now that I’ve impressed you with all my credentials, I got a different answer, using a different method.

I basically assumed the burn rate was constant and used a = F/m. For the first stage the force was 33, 361, 665N and the mass was (2812320 - 13583.6x) where x is time (all units are metric). I integrated that equation for x[sub]min[/sub] = 0 and x[sub]max[/sub]= t to get the velocity at any given point (at the end of the first burn it is 3166.27m/s), then integrated it again to find the distance traveled during the interval of 0 to 150s which is 187 806m. I did this again for the next one, except added the velocity before integrating it a second time, etc. Anyways, my answer is a final velocity of 12, 243m/s and a total distance of 6, 656km.

8

If there are no gravitational forces in play, then what effect does lightening the load have? If an object in space uses thrust to gain momentum, and we assume there is no gravity at affect the object, doesn’t the object continue to coast almost indefinitely (assuming frictionless travel)?

9

I suppose a better choice of words would have been “reducing the mass”. As you “lighten your load” but still continue being pushed along with the same force, acceleration increases because acceleration varies inversely with mass.

10

Ian Fan, I like what you did with the mass function - I may review my answer in light of that.

However, I think there’s a problem with the a = F/m part. As I stated, that interpretation of Newton’s law assumes constant mass. If the mass changes (as it does when it burns), then the real trick is force equals the time derivative of momentum, or mass times velocity.

F = d(mv)/dt You can see that if the mass is constant, that reduces to

F = m d(v)/dt, or in turn F = ma .

Attrayant, the effect of lightening the load is that there is less inertia - less resistance to change in motion. For the same amount of thrust (the engine runs at constant thrust, so each stage has a constant value), you are getting more acceleration as the mass decreases. Thus you actually go faster than the conservative assumption that mass stays constant at max value.

11

I redid the calculations for F = dp/dt and got a final velocity of 22, 364m/s and a total distance of 12, 000.2 km.

12

As Attrayant pointed out, there really is no “total distance” if you ignore gravity and friction. No matter what velocity you achieve at the end of the burn, you’ll just keeping going at the velocity forever.

Arjuna34

13

Yes, that is pretty obvious. But I assumed that enolancooper was just wondering out of curiosity as to how far the spaceship has traveled once all the fuel had been used up.

14

I remember reading that the Saturn V was the first and only vehicle capable of sending a package to another solar system. Someone figured out that if you could get a seventeen pound package entirely out of the Sun’s gravity well and drift it to Alpha Centauri in a mere couple of thousand years.