Boost Phase: Velocity at Burnout
Rocket thrust = T
Force of gravity = M*g
Drag force on rocket = 0.5*rho*Cd*A*v^2 = k*v^2
Net force on rocket = F = T  M*g  k*v^2
Newton's Second Law: F = M*a = M*(dv/dt) = T  M*g  kv^2
Collecting terms: dt = M*dv / (T  M*g  k*v^2) = (M / k)*(dv / [q^2  v^2])
where I've defined q = sqrt([T  M*g] / k)
Integrating both sides (finite integral from 0 to v) and rearranging:
t = (M / k)*(1 / [2*q])*ln([q+v] / [qv])
Simplifying a bit: 2*k*q*t / M = ln([q+v] / [qv])
Set x = 2*k*q / M and then
Solve for v:
v = q*[1  exp(x*t)] / [1 + exp(x*t)]
Boost Phase: Altitude at Burnout
Newton's Second Law Again
F = M*a = M*(dv/dt) = M*(dv/dy)*(dy/dt) = M*v*(dv/dy) = T  M*g  k*v^2
Rearranging: dy = M*v*dv / (T  M*g  k*v^2)
Integrating both sides (finite integral from 0 to v) and rearranging:
y = (M / 2*k)*ln([T  M*g  k*v^2] / [T  M*g])
Coast Phase: Distance Travelled from Velocity v to Zero
Newton's Second Law Yet Again
F = M*a = M*v*(dv/dy) =  M*g  k*v^2
Rearranging: dy = M*v*dv / ( M*g  k*v^2)
Integrating both sides: y = [+M / (2*k)]*ln([M*g + k*v^2] / [M*g])
Coast Phase: Time to Velocity Zero
Newton's Second Law in the Time form
F = M*a = M*(dv/dt) =  M*g  k*v^2
Rearranging: dt = M*dv / ( M*g  k*v^2)
Integrating both sides: t = ([M/k]/sqrt[M*g/k])*arctan(v / sqrt[M*g/k])
Simplifies (some) to: t = sqrt(M / [k*g])*arctan(v / sqrt[M*g/k])
Approximation Using Static Rocket Mass
THE rocket equation assumes a dynamic mass m(t) = m0  (dm/dt)*t, where
dm/dt is a constant. When this expression is substituted into the above
Second Law equations, they become intractable and must be solved with numerical
methods. I therefore use a static expression for the mass M of the rocket.
I will call the velocity found using the dynamic expression "vd",
and the velocity found using the static expression "vs".
Define
Vx = exhaust velocity, speed of propellant leaving rocket
mr = mass of rocket, when EMPTY
mp = mass of propellant (total)
Then we have
THE rocket equation (dynamic mass): vd = Vx * ln([mr+mp] /
mr)
The "static rocket mass" equation: vs = Vx * (mp / mr)
The static equation equivalent to my method of using average rocket mass
is:
vs = Vx * (mp / [mr + 0.5*mp]).
Then a measure of the error induced by my method is E = 1  vs / vd. Let's
suppose the extreme case (for a model rocket) that the propellant is half
the total weight of the rocket, or mp = mr = m. Then
E = 1  vs / vd = 1  [ (m / {m+0.5*m}) / ln({m+m} / m) ] = 1  [1 / {ln(2)
* 1.5}] = 0.04, or 4% error.
You can verify for yourself that for propellants that are somewhat
smaller proportions of the rocket mass, the error is much smaller. The propellant
has to exceed 67% of the total rocket mass before a 10% error is induced.
With thanks to:
Randy Culp
http://www.rocketmime.com/rockets/rckt_eqn.html
