SOME HINTS FOR PROBLEM SET
2
·
The first part of the problem set is calculating the
’s for each part of the mission. You should not get difficulties
with that. It is just like lecture 4. Just remember that you have to assume
that each 
is applied as an impulse,
which means for the equation giving 
out of 
:
The second term is zero because the spacecraft has no time
to move while the 
is applied.
·
Now you have all your 
’s, and you want to calculate the number of stages. The
bigger 
, the more stages you will need (Lecture 2).
Let’s work out the example of a three-stage rocket, with, the same exhaust velocity (4,500 m/s), maximum payload, structural and propellant mass fractions for each stage.
When I say ‘stage i’, that includes (c.f. figure)
o the classic definition of stage i,
o the upper stages, which are the payload of this stage.
For each stage:
Payload mass fraction of stage i, 
Structural mass fraction 
Propellant mass fraction 
The total mass of stage i, before it starts burning its propellant being:
Note: the payload of stage i is, by construction, the total mass of stage i+1:
(1)
The 
brought by stage i
to the final payload is, from the rocket equation:
where c is the exhaust velocity of the stage-i engine.
The gravity term is zero here because 
is applied as an impulse.
Plugging in some numbers,
For a three-stage rocket, the 
received by the payload
is:
The final payload mass fraction of the rocket, is
, using (1)
Hence,
If you need a kick of 14,400 m/s only, you can use a two-stage rocket, and the final payload mass fraction will be: 0.1*0.1=0.01
However, if you know you have to give your final payload a kick of, let’s say, 25,000 m/s, then you will have to have 4 stages, the last one being partially used (you do not have to use its maximum propellant mass fraction). Your final payload mass fraction will be smaller than 0.001 because you have added a stage.
In the Problem Set, knowing the
mass fractions, the exhaust velocities, you know how much 
is given by a stage;
you also know the required 
...