The differential equation m dv/dt = mg - kv^2 governs the velocity v of an objec

Question

The differential equation m dv/dt = mg - kv^2 governs the velocity v of an object falling near the earth's surface. Here m is the mass of the object and g is the acceleration due to gravity, and thus the mg term on the right is the force of gravity. The second term on the right is the force of air resistance. Here k = 1/2 rho AC_D, where rho is the density of air, A is the cross-sectional area perpendicular to the direction of motion, and coefficient of drag C_D depends on the shape of the object. Write a function [k] = drag constant(rho, A, CD) which calculate the constant k in the differential equation. The velocity of a falling object will approach a terminal velocity where the drag force and the gravitational force cancel. Give a formula for the terminal velocity in terms of m, g, and k. Use MATLAB's symbolic manipulation capacity to solve the differential equation assuming an initial condition v(0) = v_0. Use your solution to create a function [v, vinf] = fall velocity(t, v0, mass, rho, A, CD) that will return the velocity v and the terminal velocity. The mk 82 is a general purpose low drag bomb. The drag is kept low so that, if attached externally to an aircraft, it will minimally impede the aircraft's performance. A typical mass for an mk 82 is about 240 kg and its cross-sectional diameter is 0.27 m. Depending upon the configuration of fuzes and lugs its low velocity coefficient of drag is in the 0.1-0.3 range. Plot the velocity of an mk 82 falling straight downward for coefficients of drag C_D =0.1 and 0.3. Plot the velocity all the way to terminal velocity. Include labels and a legend. Use fzero to find when the mk 82 reaches mach 1 (about 340 m/s). Sample output: The coefficient of drag for a sphere is about 0.5. Thus for a sphere of radius 5 cm and mass 1 kg falling through air with density 1.2 kg/m^3 drag constant(1.2, pi*0.05^2, 0.5) ans = 0.0024 Taking the gravitational acceleration to be g = 9.8 m/s^2 [v, vinf] = fall velocity ([1 2 3], 0, 1, 9.8, 1.2, pi*0.50^2, 0.5) v = 9.7253 19.0181 27.5196 vinf = 64.4922

Explanation / Answer

no question displayed

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.