The forces on a rocket change dramatically
during a typical flight.
This figure shows a derivation of the change in velocity during
powered flight while accounting for the changing mass of the rocket.
During powered flight the propellants of the
propulsion system are constantly being
exhausted from the
nozzle. As a result, the
weight of the rocket is constantly
changing. In this derivation, we are going to neglect the effects of
aerodynamic lift and drag. We can add these
effects to the final answer.
Let us begin with Newton's
second law
of motion, shown in blue on the figure:
d (M u) / dt = F net
where M is the mass of the rocket, u is the velocity of the
rocket, F net is the net external force on the rocket and the
symbol d / dt denotes that this is a differential equation in
time t. The only external force which we will consider is the
thrust from the propulsion system.
On the web page describing the
specific impulse, the thrust equation is given by:
F = mdot * Veq
where mdot is the
mass flow rate, and Veq is the equivalent
exit velocity of the nozzle which is defined to be:
Veq = V exit + (p exit  p0) * Aexit / mdot
where V exit is the exit velocity, p exit is the exit
pressure, p0 is the free stream pressure,
and A exit is the exit area of the nozzle.
Veq is also related to the specific impulse Isp:
Veq = Isp * g0
where g0 is the gravitational constant.
m dot is mass flow rate and is equal to the change in the mass of
the propellants mp on board the rocket:
mdot = d mp / dt
Substituting the expression for the thrust into the motion equation gives:
d (M u) / dt = V eq * d mp / dt
d (M u) = Veq d mp
Expanding the left side of the equation:
M du + u dM = Veq d mp
Assume we are moving with the rocket, then the value of u is zero:
M du = Veq d mp
Now, if we consider the instantaneous mass of the rocket M,
the mass is composed of two main
parts,
the empty mass me and the propellant mass mp.
The empty mass does not change with
time, but the mass of propellants on board the rocket does change with time:
M(t) = me + mp (t)
Initially, the full mass of the rocket mf contains the empty mass and
all of the propellant at lift off.
At the end of the burn, the mass of the rocket contains only the
empty mass:
M initial = mf = me + mp
M final = me
The change on the mass of the rocket is equal to the change in mass of the
propellant, which is negative, since propellant mass is constantly being
ejected out of the nozzle:
dM =  d mp
If we substitute this relation into the motion equation:
M du =  Veq dM
du =  Veq dM / M
We can now integrate this equation:
delta u =  Veq ln (M)
where delta represents the change in velocity, and ln is
the symbol for the natural logarithmic
function.
The limits of integration are from the initial mass of the rocket to
the final mass of the rocket. Substituting for these values we obtain:
delta u = Veq ln (mf / me)
This equation is called the ideal rocket equation. There are
several additional forms of this equation which we list here:
Using the definition of the
propellant mass ratio MR
MR = mf / me
delta u = Veq * ln (MR)
or in terms of the specific impulse of the engine:
delta u = Isp * g0 * ln (MR)
If we have a desired delta u for a maneuver, we can invert this
equation to determine the amount of propellant required:
MR = exp (delta u / (Isp * g0) )
where exp is the exponential
function.
If you include the effects of gravity, the rocket equation becomes:
delta u = Veq ln (MR)  g0 * tb
where tb is the time for the burn.
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