In this post, we will first look at the main characteristics of rocket flight, and then examine the feasibility of launching a satellite as the payload of a rocket into an orbit above the earth.
A rocket accelerates itself by ejecting part of its mass with high velocity.
Fig. 1 shows a moving rocket. At time , the mass leaves the rocket in opposite direction. As a result, the rocket is being propelled away with an increased speed.
– the mass of rocket at time
– the velocity of rocket at time
– the magnitude of
– the velocity of ejected mass at
– the magnitude of
– the magnitude of ‘s velocity relative to the rocket when it is ejected. It is time invariant.
From Fig. 1, we have
and most notably, the relationship between and (see “A Thought Experiment on Velocities”):
It follows that
momentum at time :
momentum at time : .
Consequently, change of momentum in is .
Apply Newton’s second law of motion to the whole system,
where is the sum of external forces acting on the system.
To get an overall picture of the rocket flight, we will neglect all external forces.
Without any external force, . Therefore
That fact that in (1) are positive quantities shows as the rocket loses mass (), its velocity increases ()
Integrate (1) with respect to ,
where is the constant of integration.
At where is the initial rocket mass (liquid or solid fuel + casing and instruments, exclude payload) and the payload.
It means .
As a result,
Since is divided into two parts, the initial fuel mass , and the casing and instruments of mass , can be written as
When all the fuel has burnt out at ,
By (2), the rocket’s final speed at
In other words,
Hence, the final speed depends on three parameters
Using these values, (3) gives
This is an upper estimate to the typical final speed a single stage rocket can give to its payload. Neglected external forces such as gravity and air resistance would have reduced this speed.
With (4) in mind, let’s find out whether a satellite can be put into earth’s orbit as the payload of a single stage rocket.
We need to determine the speed that a satellite needs to have in order to stay in a circular orbit of height above the earth, as illustrated in Fig. 2.
By Newton’s inverse square law of attraction, The gravitational pull on satellite with mass is
where universal gravitational constant , the earth’s mass , and the earth’s radius .
For a satellite to circle around the earth with a velocity of magnitude , it must be true that
On a typical orbit, above earth’s surface,
This is far in excess of (4), the value obtained from a single stage rocket.
The implication is that a typical single stage rocket cannot serve as the launching vehicle of satellite orbiting around earth.
We will turn to multi-stage rocket in “Viva Rocketry! Part 2“.