Final answer:
To calculate the total mass of the rocket at lift-off, we need to consider the masses of the payload, structure, and fuel for both stages. By using the specific impulse (Isp) and exhaust velocity (Ve) values for each stage, we can calculate the initial and final masses of each stage. Adding the masses of all components gives us the total mass of the vehicle at lift-off, which is 22,664 kg.
Step-by-step explanation:
To calculate the total mass of the vehicle at lift-off, we need to consider the masses of the payload, structure, and fuel for both stages of the rocket. Let's calculate the masses for each stage:
Stage 1:
- Exhaust velocity (Ve1) = Isp1 * g = 390 * 9.8 = 3822 m/s
- Initial mass (mi1) = Payload mass + Structure mass stage 1 = 150 + 9004 = 9154 kg
- Final mass (mf1) = mi1 / e^(ΔV1 / Ve1) = 9154 / e^(3900 / 3822) = 9154 / e^(0.988) = 9154 / 2.68 = 3414 kg
- Fuel mass stage 1 (m1) = mi1 - mf1 = 9154 - 3414 = 5740 kg
Stage 2:
- Exhaust velocity (Ve2) = Isp2 * g = 430 * 9.8 = 4214 m/s
- Initial mass (mi2) = Structure mass stage 2 + Fuel mass stage 1 = 2500 + 5740 = 8240 kg
- Final mass (mf2) = mi2 / e^(ΔV2 / Ve2) = 8240 / e^(4300 / 4214) = 8240 / e^(1.019) = 8240 / 2.77 = 2970 kg
- Fuel mass stage 2 (m2) = mi2 - mf2 = 8240 - 2970 = 5270 kg
Total mass at lift-off = Payload mass + Structure mass stage 1 + Structure mass stage 2 + Fuel mass stage 1 + Fuel mass stage 2 = 150 + 9004 + 2500 + 5740 + 5270 = 22,664 kg