Question

what are the equations that can be used to calculate the rocket launching

Answer 1

Altitude is a very minor consideration when launching a rocket.

The main thing is velocity, which has to be high enough to first enter orbit at about 5 miles per second, and then break orbit with an additional 2 miles per second.

i,e You need a delta V (total change in speed) of a little more than 11 km/sec.

you need to know what exhaust velocity your engine will produce.

The size of the rocket itself includes a lot of factors

ou need a high thrust-to-weight ratio initially just to achieve partial orbit. After that specific impulse becomes more important, which determines how much weight is used to achieve a change in velocity

you can calculate the mass fraction needed to get that speed using the rocket equation.

The mass fraction is the total mass of the empty rocket (with payload but no fuel) to the fully fueled rocket

you decide how many stages to use, because you can multiply the mass fraction of each stage to get the total mass fraction of the rocket.

you need to ask your engineering team how small they can make a rocket with that mass fraction. In theory there is no lower limit, but in reality engineers can only make rocket engine parts that still work so small and no smaller.

To escape from earth, you have to expend at least the total gravitational potential energy from the point you start at.

From surface (R = 6400km, 10kg payload)

U = -GMm/R

U = -6.230 x 10^8 J

From altitude (R = 6426km, 10kg payload)

U = -6.206 x 10^8 J