$1\mathrm{.}21$ As noted in Sec. $1\mathrm{.}4,$ a fundamental representation of the drag force, which assumes turbulent conditions, can be formulated as

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

where $F_d=$ the drag force $(N),=$ fluid density $(k\frac{g}{m^3}),A=$ the frontal area of the object on a plane perpendicular to the direction of motion $(m^2),v=$ velocity $(\frac{m}{s}),$ and $C_d=$ a dimensionless drag coefficient.

$($a$)$ Write the pair of differential equations for velocity and position $($see Prob. $1\mathrm{.}19)$ to describe the vertical motion of a sphere with diameter, $d(m),$ and a density of ${}_s(\frac{kg}{m^3}).$ 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 $t=2s$ to compute the position and velocity of a sphere over the first $14$ seconds. Employ the following parameters in your calculation: $d=120cm,=1\mathrm{.}3k\frac{g}{m^3},{}_s=2700k\frac{g}{m^3},$ and $C_d=0\mathrm{.}47.$ Assume that the sphere has the initial conditions: $x(0)=100m$ and $v(0)=-40\frac{m}{s}.$

$($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

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ground.

$($d$)$ Compute the value for the bulk second$-$order drag coefficient, $c_d(k\frac{g}{m}).$ Note that the bulk second$-$order drag coefficient is the term in the final differential equation for velocity that multiplies the term $v|v|.$