Assume you have a rocket that requires a change in velocity of DeltaV=9.6km/s . There are two options for the propulsion system --- 1. chemical and 2. electric --- each with a different specific impulse.
Using the setting of g0 =9.81m/s^2 to calculate the propellant fraction required to achieve the necessary DeltaV for each of propulsion system answer the following two questons.
Question 1
First, consider a cryogenic chemical propulsion system with Isp Isp=450s . Enter the required propellant fraction as a proportion with at least 2 decimal places
Question 2
Next, consider an electric propulsion system with Isp Isp2000s . Enter the required propellant fraction as a proportion with at least 2 decimal places
0.82
To calculate the propellant fraction required for each propulsion system, we can use the Tsiolkovsky rocket equation:
ΔV = Isp * ln(m0/mf)
Where:
ΔV = Change in velocity
Isp = Specific impulse
m0 = Initial mass (rocket + propellant)
mf = Final mass (rocket)
For Question 1, using a cryogenic chemical propulsion system with Isp = 450s:
Let's assume the rocket mass without propellant (m0) is M kg. We can calculate the required propellant fraction as follows:
ΔV = Isp * ln(m0/mf)
9.6 km/s = 450s * ln(M/mf)
Dividing both sides by 450s:
0.0213 km = ln(M/mf)
Now, let's convert km to meters:
0.0213 km = 21.3 m
Therefore, the required propellant fraction for the cryogenic chemical propulsion system is 21.3/100 = 0.213 (2 decimal places).
For Question 2, using an electric propulsion system with Isp = 2000s:
Following the same calculation, we have:
ΔV = Isp * ln(m0/mf)
9.6 km/s = 2000s * ln(M/mf)
Dividing both sides by 2000s:
0.0048 km = ln(M/mf)
Converting km to meters:
0.0048 km = 4.8 m
Therefore, the required propellant fraction for the electric propulsion system is 4.8/100 = 0.048 (2 decimal places).
To calculate the propellant fraction required for each propulsion system, we need to use the rocket equation:
ΔV = Isp * g0 * ln(m0/mf)
where:
ΔV = change in velocity (9.6 km/s in this case)
Isp = specific impulse of the propulsion system
g0 = standard gravity (9.81 m/s^2)
m0 = initial mass of the rocket (including propellant)
mf = final mass of the rocket (after expelling propellant)
Let's solve each question using the given information:
Question 1:
For the cryogenic chemical propulsion system with Isp = 450 s:
ΔV = 450 * 9.81 * ln(m0/mf)
We need to solve for the propellant fraction (mf/m0). Rearranging the equation:
mf/m0 = e^(-ΔV / (450 * 9.81))
Now, substitute the given ΔV = 9.6 km/s:
mf/m0 = e^(-9600 / (450 * 9.81))
Calculating this using a calculator or software, we find mf/m0 ≈ 0.2978. Therefore, the required propellant fraction is approximately 0.2978 for the cryogenic chemical propulsion system.
Question 2:
For the electric propulsion system with Isp = 2000 s:
ΔV = 2000 * 9.81 * ln(m0/mf)
Again, we need to solve for the propellant fraction (mf/m0). Rearranging the equation:
mf/m0 = e^(-ΔV / (2000 * 9.81))
Substituting the given ΔV = 9.6 km/s:
mf/m0 = e^(-9600 / (2000 * 9.81))
Calculating this using a calculator or software, we find mf/m0 ≈ 0.0931. Therefore, the required propellant fraction is approximately 0.0931 for the electric propulsion system.