The drag force on an object moving through a fluid, denoted by $F_{\text{d}\text{r}\text{a}\text{g}},$ is known to be directly

proportional to the square of the speed relative to the fluid:

$F_{\text{d}\text{r}\text{a}\text{g}}=k{v}^2,$

where $k$ is a dimensional constant of proportionality. Assume that a tiny particle is released

from rest at the origin and take the $y-$axis pointing downward in the direction of motion with

the origin at the initial position. In this problem, you will be asked to derive a formula for the

equation governing the motion of the object and solve this equation to answer the following

questions. Be sure to verify that your formula is dimensionally consistent.

$($a$)$ Draw a free$-$body diagram for this situation and use your free$-$body diagram with

Newton's second law to write down the governing equation for this situation.

$($b$)$ Derive an expression for the velocity as a function of time $v=v(t).$

$($c$)$ Integrate the expression for $v=v(t)$ to find the position of the particle as a function

of time $y=y(t).$

$($d$)$ Determine the speed of the terminal velocity $v_{\text{t}\text{e}\text{r}\text{m}}.$