Say a ball of mass M and diameter D is in free-fall from a very high velocity altitude. The governing differential equation for the free-fall velocity V(t) is:
M*(dV/dt)=Mg-F(Drag); V(o)=0 (1)
where gravity g= 9.81 m/sec^2 and F(drag) is the viscous force on the air.
(a.) Fluid Mechanics Solution: From Fluid Mechanics, the drag force due to th air turns out to be: F(Drag)=C*rho(air)V^2*Area*(0.5) (2)
Lets take: density of the air (rho(air)) to be 1 kg/m^3, the drag coefficient to be C=1, and the surface area of the ball to be Area= pi*D^2.
1.) What is the terminal velocity, V(t) of the ball?
2.) Solve the differential equation (1) for V(t) using the fluid-mechanical form of the drag force equation (2).
b.) System Dyamics Solution: System dynamics methods usually entail posing the active forces in a linear form, like:
F(Drag)=bV (3)
One of the core issues for the system-dynamics modeler is to pick a physically-justified value for the constraints like b.
1.) Solve for V(t) using the dynamics system for of F(Drag), equation (3), on the right-hand side of the equation (1).
2.) Looking at your answer, what must you set the constant b to, in order that the system-dynamics expression produces the correct (fluid-dynamics) value for the free-fall terminal velocity?
c.) MATLAB Plots:
1.) Create matlab functions to plot both your system-dynamics solution for V(t) and the fluid-dynamics solution for V(t) on the same plot.
2.) Please provide the MatLab code and sample output plots with labeled axis', and a legend for each solution.