Via a google search, here are some of the formulas you can use to calculate different portions of the trajectory of your model rocket. Unless you are using a laser ranging tool, accuracy will be based on estimates made at each point. To help reduce the margin of error we recommend a specific individuals to be in charge of timing and focusing on a specific aspect of the launch. This helps stick closer to the 10% margin of error.
This HTML version of the post below helps greatly when it comes to the formulas. As the article below states, the formulas look intimidating at first but they are easy once you run them a few times.
Also see the .PDF file to download. One of the sources for the information is on the link below.
http://www.rocketmime.com/rockets/rckt_eqn.html
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] / [q-v])
God this makes me wish I had something better than html for these equations.
Anyway, - Simplifying a bit: 2*k*q*t / M = ln([q+v] / [q-v])
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.
