Rocket Science

1
Rocket Science

• Joseph Thomason and Gray Sanborn

2
Background

• Most rockets used to launch satellites and that
  perform other missions use three stages.
• These stages use all fuel, and then jettison to
  reduce the mass of the rocket.
• We need to determine the individual mass of each
  stage to minimize the total mass of the rocket,
  yet allow it to reach a desired velocity.

3
Setup

• Change in velocity is given as
• ?V - c ln 1 (1-S)Mr/(P Mr)
• Mr mass of rocket at an individual stage
  including fuel.
• P mass of payload (including other stages
  remaining)
• S structural constant
• c speed of exhaust relative to rocket
  (constant)
• Also given that A is the payload of the rocket
  being delivered.

4
Final Velocity

• We need to show that final velocity, Vf, is given
  by
• Vf c ln (M1 M2 M3 A)/(SM1 M2 M3
  A) ln (M2 M3 A)/(SM2 M3 A) ln (M3
  A)/(SM3 A)

5
Proof for Final Velocity

• ?V - c ln 1 (1-S)Mr/(P Mr)
• At stage 3, Mr M3, P A
• V3 -c ln 1 (1-S) M3/(M3 A)
• V3 -c ln(M3 A M3 SM3)/(M3 A)
• V3 -c ln(SM3 A)/(M3 A)
• V3 c ln(M3 A)/(SM3 A)
• At stage 2 , Mr M2, P M3 A
• V2-c ln 1 (1-S) M2/(M2 M3 A)
• V2 -c ln(M2 M3 A M2 SM2)/(M2 M3
  A)
• V2 -c ln(SM2 M3 A)/(M2 M3 A)
• V2 c ln(M2 M3 A)/(SM2 M3 A)

6
Proof for Final Velocity ctd.

• At stage 1 , Mr M1, P M2 M3 A
• V1-c ln 1 (1-S) M1/(M1 M2 M3 A)
• V1 -c ln(M1 M2 M3 A M1 SM1)/(M1 M2
  M3 A)
• V1 -c ln(SM2 M3 A)/(M1 M2 M3 A)
• V1 c ln(M1 M2 M3 A)/(SM2 M3 A)
• Since Vf V1 V2 V3, Vf c ln (M1 M2
  M3 A)/(SM1 M2 M3 A) ln (M2 M3
  A)/(SM2 M3 A) ln (M3 A)/(SM3 A)

7
Minimizing the Total Mass

• In order to minimize the mass, we need to use
  this final velocity function as a constraint for
  a Lagrange multiplier.
• However, we will simplify this function by using
  variables Ni such that Vfcln N1 ln N2 ln N3

8
Proof 2

• We need to prove that
• (M1 M2 M3 A)/(M2 M3 A)
  (1-S)N1/(1-SN1)
• N1 (M1 M2 M3 A)/(SM1 M2 M3 A)
• (1-S)(M1 M2 M3 A/SM1 M2 M3 A)/(1
  S(M1 M2 M3 A)/(SM1 M2 M3 A))
• (M1 M2 M3 A SM1 SM2 SM3 SA)/(SM1
  M2 M3 A)/(SM1 M2 M3 A SM1 SM2
  SM3 SA)/(SM1 M2 M3 A)
• M1 M2 M3 A SM1 SM2 SM3 SA/M2 M3
  A SM2 SM3 SA
• (1-S)/(1-S)(M1 M2 M3 A)/(M2 M3 A)

9
Proof 2 ctd.

• (M2 M3 A)/(M3 A) (1-S)N2/(1-SN2)
• N2 (M2 M3 A)/(SM2 M3 A)
• (M2 M3 A SM2 SM3 SA)/(SM2 M3
  A)/(SM2 M3 A SM2 SM3 SA)/(SM2 M3
  A)
• (M2 M3 A SM2 SM3 SA)/(M3 A SM3 SA)
• (1-S)/(1-S)(M2 M3 A)/(M3 A)
• (M3 A)/A (1-S)N3/(1-SN3)
• N3 (M3 A)/(SM3 A)
• (M3 A SM3 SA)/(SM3 A)/(SM3 A SM3
  SA)/(SM3 A)
• (M3 A SM3 SA)/(A-SA)
• (1-S)/(1-S)(M3 A)/A

10
Final Conclusion to proof 2

• (M A)/A (1-S)3(N1N2N3)/(1-SN1)(1-SN2)(1-SN
  3)
• Since all three are multiplied together, you have
  N1N2N3.
• (M1 M2 M3 A)/(M2 M3 A)(M2 M3
  A)/(M3 A)(M3 A)/(A)
• By multiplying, you get (M1 M2 M3 A)/A
  (MA)/A

11
Using Lagrange

• f c ln ((M A)/A)
• (MA)/A) (1-S)3N1N2N3/(1-SN1)(1-SN2)(1-SN3)
• f 3c ln (1-S) c ln N1 c ln N2 c ln N3 c
  ln (1-SN1) c ln (1-SN2) c ln (1-SN3)
• Gradient of f(N1, N2, N3) lt(c/N1(1-SN1)),
  (c/N2(1-SN2)), (c/N3(1-SN3))gt
• Constraint, g c ln N1 ln N2 ln N3 Vf
  0
• Gradient of g(N1, N2, N3) lt(c/N1), (c/N2),
  (c/N3)gt

12
Lagrange contd.

• We have the following equations from this.
• c/N1(1-SN1) ?c/N1
• c/N2(1-SN2) ?c/N2
• c/N3(1-SN3) ?c/N3
• c ln N1 ln N2 ln N3 Vf
• We can see that in the first three equations
  that
• ? 1/(1-SN1)
• ? 1/(1-SN2)
• ? 1/(1-SN3)
• from which we can deduce that N1 N2 N3 when
  mass is minimized.

13
Using N1 N2 N3 to find minimum

• With the constraint Vf c ln N1 ln N2 ln
  N3, we can change this to Vf 3c ln N3 to find
  the mass, M3
• Vf/3c ln (M3 A)/(SM3A)
• For reference, E e(Vf/3c)
• E (M3 A)/(SM3 A)
• (SM3 A)(E) M3 A
• E(SM3 A) M3 A 0
• M3(SE-1) A(E-1) 0
• M3 -A(E-1)/(SE-1)

14
Minimum part 2

• In a likewise manner, (in order to save time), we
  obtain
• M2 -A(E-1)(SE-E)/(SE-1)2
• M1 (E-1)/(SE-1)3A(E-1)(SE-E)-(SE-1)(E-1)
  (SE-1)2
• M M1 M2 M3

15
Application

• We want to put a rocket above the earths
  surface.
• Vf 17,500 mph
• S 0.2
• c 6000 mph
• E e(Vf/3c) 2.644
• Using the previous values of M1, M2, and M3, to
  obtain minimum masses for this rocket, we have
  obtained that
• M3 3.488A
• M2 15.656A
• M1 38.958A
• M 58.103A

16
Application part 2

• Same rocket wants to put a payload of 500lbs into
  deep space at velocity of 24,700 mph
• Vf 24,700 mph
• S 0.2
• C 6000 mph
• A 500 lbs
• E e(Vf/3c) 3.944
• Using our determinations for M1, M2, and M3, we
  have
• M3 6970.644 lbs
• M2 104,150.412 lbs
• M1 1,347,840.562 lbs
• M 1,458,961.618 lbs.