As noted in Sec. 1.4, a fundamental representation of the drag force, which assumes turbul...

As noted in Sec. 1.4, a 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), ρ = fluid density (kg/m³), A = the frontal area of the object on a plane perpendicular to the direction of motion (m²), υ = velocity (m/s), and C_d = a dimensionless drag coefficient.

(a) Write the pair of differential equations for velocity and position (see Prob. 1.19) to describe the vertical motion of a sphere with diameter, d (m), and a density of ρ_s (kg/m³). 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 seconds. Employ the following parameters in your calculation: d = 120 cm, ρ = 1.3 kg/m³, ρ_s = 2700 kg/m³, and C_d = 0.47. Assume that the sphere has the initial conditions: x(0) = 100 m and υ(0) = −40 m/s.

---

(c) Develop a plot of your results (i.e., y and υ 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 the bulk second-order drag coefficient is the term in the final differential equation for velocity that multiplies the term υ |υ|.