Modelrockets.NLModelraket Hoogte Berekeningen - Formules

 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]) 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. With thanks to: Randy Culp http://www.rocketmime.com/rockets/rckt_eqn.html (Auto close)