Answer: Part 1: Propellant Fraction (MR) = 8.76
Part 2: Propellant Fraction (MR) = 1.63
Explanation: The Ideal Rocket Equation is given by:
Δv = ![v_{ex}.ln(\frac{m_{f}}{m_{e}} )](https://tex.z-dn.net/?f=v_%7Bex%7D.ln%28%5Cfrac%7Bm_%7Bf%7D%7D%7Bm_%7Be%7D%7D%20%29)
Where:
is relationship between exhaust velocity and specific impulse
is the porpellant fraction, also written as MR.
The relationship
is: ![v_{ex} = g_{0}.Isp](https://tex.z-dn.net/?f=v_%7Bex%7D%20%3D%20g_%7B0%7D.Isp)
To determine the fraction:
Δv = ![v_{ex}.ln(\frac{m_{f}}{m_{e}} )](https://tex.z-dn.net/?f=v_%7Bex%7D.ln%28%5Cfrac%7Bm_%7Bf%7D%7D%7Bm_%7Be%7D%7D%20%29)
![ln(MR) = \frac{v}{v_{ex}}](https://tex.z-dn.net/?f=ln%28MR%29%20%3D%20%5Cfrac%7Bv%7D%7Bv_%7Bex%7D%7D)
Knowing that change in velocity is Δv = 9.6km/s and
= 9.81m/s²
<u>Note:</u> Velocity and gravity have different measures, so to cancel them out, transform km in m by multiplying velocity by 10³.
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<u>Part 1</u>: Isp = 450s
![ln(MR) = \frac{v}{v_{ex}}](https://tex.z-dn.net/?f=ln%28MR%29%20%3D%20%5Cfrac%7Bv%7D%7Bv_%7Bex%7D%7D)
ln(MR) = ![\frac{9.6.10^{3}}{9.81.450}](https://tex.z-dn.net/?f=%5Cfrac%7B9.6.10%5E%7B3%7D%7D%7B9.81.450%7D)
ln (MR) = 2.17
MR = ![e^{2.17}](https://tex.z-dn.net/?f=e%5E%7B2.17%7D)
MR = 8.76
<u>Part 2:</u> Isp = 2000s
![ln(MR) = \frac{v}{v_{ex}}](https://tex.z-dn.net/?f=ln%28MR%29%20%3D%20%5Cfrac%7Bv%7D%7Bv_%7Bex%7D%7D)
ln (MR) = ![\frac{9.6.10^{3}}{9.81.2.10^{3}}](https://tex.z-dn.net/?f=%5Cfrac%7B9.6.10%5E%7B3%7D%7D%7B9.81.2.10%5E%7B3%7D%7D)
ln (MR) = 0.49
MR = ![e^{0.49}](https://tex.z-dn.net/?f=e%5E%7B0.49%7D)
MR = 1.63