...
where v_2 - v_1 = v_2 * cos(Lambdaλ)
where c_p = 1004.5 for air and Lambda is the sweep angle of the fins
Assuming the rocket reaches a target of 30K ft AGL:
T_static = 59F - .00356(F/ft) * h_{max_v} (ft)
Therefore,
T_stag = (59F - .00356(F/ft) * h_{max_v} (ft)) + (4.9776*10^{-4}) * (v_2 * cos(λ))^2
v_2 = Mach 2 = 2*343 (m/s) = 686 (m/s)
T_stag = (59F - .00356(F/ft) * h_{max_v} (ft)) + (4.9776*10^{-4}) * (686 (m/s) * cos(λ))^2
Note that the first half of the sum is in degrees F whereas the second half of the sum will be in degrees K.
Dynamic Pressure
Dynamic pressure (q) represents the aerodynamic pressure exerted on a vehicle in motion through a fluid (air). It is defined by the equation:
q = (1/2)*ρ*v^2
where:
ρ = density of air
v = vehicle velocity
As a rocket ascends through the atmosphere, its velocity increases, and the air density decreases. The point at which these two factors result in the greatest aerodynamic load is known as Max Q—the maximum dynamic pressure. This moment represents the peak mechanical stress experienced by the rocket due to aerodynamic forces.
Structural integrity of the rocket must be maximized to withstand the peak load at Max Q.
Dynamic Stability
Dynamic Stability is how systems respond to disruptions over time. Defined by:
d^2x/dt^2+2𝞯⍵(n)(dx/dt)+⍵(n)^2x=0
In which:
x: displacement
𝞯: damping ratio = (c/c_c): actual damping/ critical damping
⍵(n): natural frequency