Free-fall under gravity with air resistance may be modelled by various different

Question

Free-fall under gravity with air resistance may be modelled by various differential equations. For this problem, we consider a model with quadratic drag, i.e., the drag force on a falling body (skydiver in our case) is proportional to the square of the downward velocity, represented by v m s-1 at time t seconds after drop. Along with initial condition v - 0, we have Equation (1), in which m kg is the skydiver mass, g ms-2 is acceleration due to gravity and k units is the so-called drag coefficient. Determine the units of the drag coefficient k. By making the changes of variable v = alpha V, and t = beta T, show that for suitable choices of the parameters pi > 0 and beta > 0, Equation (1) may be transformed into Equation (2). Note that both V and T are non-dimensional quantities. By using the method of separation of variables, or otherwise, find the general solution of Equation (2) in terms of one instance of a single, common transcendental function. Show your working carefully Given v = 0 when t = 0, compute the particular solution of Equation (2) for V as a function of T. Use appropriate computer software to plot the solution of the previous part for 0 T 5. Include hardcopy with your submission. For the particular solution found in part (d), algebraically determine the value of T (to 4 significant decimal digits) when the falling body reaches 90% of terminal velocity.

Explanation / Answer

A. The Drag Coecient
[Imai (1951)] gives an expression for the drag force in terms of an arbitrary closed circular contour of radius
r0 surrounding the circular cross-section of the cylinder. Converting this expression to polar coordinates,
and dening the Reynolds number as ? ? Re ? U?L?f/

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

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