$0\mathrm{.}2$ Model $2$: Drag force proportional to squared speed

Now suppose that force of wind resistance varies as $v^2$ :

$F_{\text{a}\text{i}\text{r}}=-v^2.$

Compared to the previous model, this means the force increases much more rapidly for large velocities. Doubling the speed will quadruple the drag force, tripling the speed will multiply the drag force ninefold, etc.

When in flight, the drag force is the only force acting on the bullet in the horizontal direction. Thus Newton's $2$nd law reads

$-v^2=ma$

rearranged into standard format for a differential equation in $v(t)$ it reads

$v^{\text{'}}(t)+\frac{}{m}(v(t){)}^2=0.$

This is also first order linear, and separable, so it can be solved as follows:

$\frac{v^{\text{'}}(t)}{(v(t){)}^2}=-\frac{}{m}=>{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{v^{\text{'}}(t)dt}{(v(t){)}^2}={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}-\frac{}{m}dt=>{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dv}{v^2}=-\frac{}{m}t+c$

Exercise $5.$ Integrate the left hand side of this last expression and use the initial condition $v(0)=v_0$ to solve for $c$ to obtain an expression for $v(t),$ the speed of the bullet $t$ seconds after firing.

Check that your solution is valid by computing $v^{\text{'}}(t)$ and plugging it and $v(t)$ into differential equation $(2)$ to confirm that it is a true equation for all $t.$

Exercise $6.$ Integrate the expression for $v(t)$ with respect to $t$ and use the initial condition $x(0)=0$ to obtain an expression for $x(t).$

Exercise $7.$ Examine the expressions you've found for $v(t)$ and $x(t)$ to find a relationship between $v$ and $x.$ In other words, determine how the speed $v$ of the bullet depends on the distance $x$ it has traveled downrange.

If you complete the preceding exercise correctly, you'll discover that the predicted relationship between bullet speed and distance downrange is that speed decays exponentially with distance downrange: $v=v_0{e}^{-rx},$ where the exponential decay rate $r$ is related to the mass of the bullet and the aerodynamic$/$drag coefficient $.$

Exercise $8.$ Fit an appropriate "trendline" to the speed vs distance data from the box of an ammunition of your choosing. Use the fitted exponential decay rate to compute a value of $,$ the drag force $($or aerodynamic$)$ coefficient for the model where drag force is proportional to the square of the speed.