As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity.

Question:

As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity. A more fundamental representation of the drag force, which assumes turbulent conditions (i.e., a high Reynolds number), can be formulated as

where F_d = the drag force (N), r = fluid density (kg/m³), A = the frontal area of the object on a plane perpendicular to the direction of motion (m²), ν = velocity (m/s), and C_d = a dimensionless drag coefficient.

(a) Write the pair of differential equations for velocity and position (see Prob. 1.18) to describe the vertical motion of a sphere with diameter d (m) and a density of p_s (kg/km³). The differential equation for velocity should be written as a function of the sphere’s diameter.

(b) Use Euler’s method with a step size of Δt = 2 s to compute the position and velocity of a sphere over the first 14 s. Employ the following parameters in your calculation: d = 120 cm, p = 1.3 kg/m³, p_s = 2700 kg/m3, and C_d = 0.47. Assume that the sphere has the initial conditions: x(0) = 100 m and ν(0) = –40 m/s.

(c) Develop a plot of your results (i.e., y and ν versus t) and use it to graphically estimate when the sphere would hit the ground.

(d) Compute the value for the bulk second-order drag coefficient C_d’ (kg/m). Note that, as described in Prob. 1.3, the bulk second order drag coefficient is the term in the final differential equation for velocity that multiplies the term ν | ν |.