1. Package arrives the same day it was ordered
2. Package arrives in same condition it departed the warehouse
3. Amazon specifies
a minimum delivery window of 9 hours for same-day service.
1. Earth-Moon distance: 384,400 km
2. Moon orbital velocity ~= 1.022 km/s
3. Gravity of Earth (surface) 9.8 m/s/s
4. Gravity of Earth (lunar orbit) 271.72 cm/s/s
If we assume the rocket is launched within 30 minutes of payment, and the package arrives at its final destination within 30 minutes of lunar touchdown, we get an 8-hour trip.
To make this trip, our rocket must have an average velocity of:
384400 / 8 = 48050
48050 / 60^2 = 13.34722 km/s
If we split the time in half, and use 4 hours to accelerate toward orbital altitude, and another 4 hours for braking and orbital insertion (this is the easiest way for me to break down the calculations. Would love to see a more efficient solution) then our peak velocity while achieving orbital altitude is 26.694 km/s
So, our bare-bones Delta-V requirement (ignoring gravity) appears to be:
--26.694 km/s up
--26.694 km/s brake
+1.022 km/s match speed with moon
= 54.41 km/s + whatever gravity adds
Since our window for delivery is so short, we can't really take advantage of efficient orbital paths, we basically have to go screaming straight up against gravity the whole way (at least until we are close enough for lunar gravity to be dominant). Our average gravitational resistance is 5.036 m/s/s. Applied over our 8-hour trip, and gravity adds:
5.036 * 8 * 60^2 = 145.037 km/s to our delta-V requirements, bringing us to a total of 199.447 km/s of delta-V required to achieve same-day delivery.
using the Tsiolkovsky rocket equation, we can estimate the fuel required to get us this Delta-V. I will assume the following variables:
Delta-V: 199.447 km/s (from calculations above)
exhaust velocity 4.5 km/s (wiki value for bipropellants)
final mass: 5,000 kg (1/3 mass of lunar lander from apollo missions. Amazon now uses remote-piloted lunar rockets)
re-arranging the Tsiolkovsky rocket equation to solve for initial mass is beyond my level of mathematical education (how do i distributive property logarithms?), So i plugged the numbers into Wolfram Alpha and got an initial mass of 8.863e22 kg, about 1.5% the mass of the earth. unfortunately, this much mass is impossible to accelerate that quickly with such a slow exhaust stream...
If we build a rocket with an exhaust velocity of 10 km/s, we only need an initial mass of 2.426 trillion kilograms. If we get our exhaust velocity up to 50 km/s, we now only need 273 metric tons of rocket, and our hit time becomes achievable.
Of course, i'm not entirely certain what would happen if we pointed a 50 km/s exhaust plume at our planet, but i feel like the launch site would not survive.