Chapter 1, Problem 21P

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

$F_d=-\frac{1}{2}\rho A{C}_{d}v\left|v\right|$

where $F_d=$ the drag force $\left(\text{N}\right),r=$ fluid density $\left({\text{kg/m}}^{\text{3}}\right)$, $A=$ the frontal area of the object on a plane perpendicular to the direction of motion $\left({\text{m}}^{\text{2}}\right),y=$ 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 $d\left(\text{m}\right)$ and a density of ${\rho }_s\left({\text{kg/km}}^{\text{3}}\right)$. 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 $\Delta t=2$ stocompute the position and velocity of a sphere over the first 14 s. Employ the following parameters in your calculation: $d=120$ cm, $\rho =1.3{\text{ kg/m}}^{\text{3}},{\rho }_s=2700{\text{ kg/m}}^{\text{3}}$, and $C_d=0.47$. Assume that the sphere has the initial conditions: $x\left(0\right)=100\text{ m and }v\left(0\right)=–40\text{ m/s}$.

(c) Develop a plot of your results (i.e., y and v 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 $c_d\text{' }\left(\text{kg/m}\right)$. 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 $v\left|v\right|$.