We are asked to determine the velocity of a rain drop if it falls from 4 km.
To do that we will use the following formula:
![2ah=v_f^2-v_0^2](https://tex.z-dn.net/?f=2ah%3Dv_f%5E2-v_0%5E2)
Where:
![\begin{gathered} a=\text{ acceleration} \\ h=\text{ height} \\ v_f,v_0=\text{ final and initial velocity} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20a%3D%5Ctext%7B%20acceleration%7D%20%5C%5C%20h%3D%5Ctext%7B%20height%7D%20%5C%5C%20v_f%2Cv_0%3D%5Ctext%7B%20final%20and%20initial%20velocity%7D%20%5Cend%7Bgathered%7D)
If we assume the initial velocity to be 0 we get:
![2ah=v_f^2](https://tex.z-dn.net/?f=2ah%3Dv_f%5E2)
The acceleration is the acceleration due to gravity:
![2gh=v_f^2](https://tex.z-dn.net/?f=2gh%3Dv_f%5E2)
Now, we take the square root to both sides:
![\sqrt{2gh}=v_f](https://tex.z-dn.net/?f=%5Csqrt%7B2gh%7D%3Dv_f)
Now, we substitute the values:
![\sqrt{2(9.8\frac{m}{s^2})(4000m)}=v_f](https://tex.z-dn.net/?f=%5Csqrt%7B2%289.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%29%284000m%29%7D%3Dv_f)
solving the operations:
![280\frac{m}{s}=v](https://tex.z-dn.net/?f=280%5Cfrac%7Bm%7D%7Bs%7D%3Dv)
Therefore, the velocity without air drag is 280 m/s.
Part B. we are asked to determine the velocity if there is air drag. To do that we will use the following formula:
![F_d=\frac{1}{2}C\rho_{air}Av^2](https://tex.z-dn.net/?f=F_d%3D%5Cfrac%7B1%7D%7B2%7DC%5Crho_%7Bair%7DAv%5E2)
Where:
![\begin{gathered} F_d=drag\text{ force} \\ C=\text{ constant} \\ \rho_{air}=\text{ density of air} \\ A=\text{ area} \\ v=\text{ velocity} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20F_d%3Ddrag%5Ctext%7B%20force%7D%20%5C%5C%20C%3D%5Ctext%7B%20constant%7D%20%5C%5C%20%5Crho_%7Bair%7D%3D%5Ctext%7B%20density%20of%20air%7D%20%5C%5C%20A%3D%5Ctext%7B%20area%7D%20%5C%5C%20v%3D%5Ctext%7B%20velocity%7D%20%5Cend%7Bgathered%7D)
We need to determine the drag force. To do that we will use the following free-body diagram:
Since the velocity that the raindrop reaches is the terminal velocity and its a constant velocity this means that the acceleration is zero and therefore the forces are balanced:
![F_d=mg](https://tex.z-dn.net/?f=F_d%3Dmg)
Now, we determine the mass of the raindrop using the following formula:
![m=\rho_{water}V](https://tex.z-dn.net/?f=m%3D%5Crho_%7Bwater%7DV)
Where:
![\begin{gathered} \rho_{water}=\text{ density of water} \\ V=\text{ volume} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Crho_%7Bwater%7D%3D%5Ctext%7B%20density%20of%20water%7D%20%5C%5C%20V%3D%5Ctext%7B%20volume%7D%20%5Cend%7Bgathered%7D)
The volume is the volume of a sphere, therefore:
![m=\rho_{water}(\frac{4}{3}\pi r^3)](https://tex.z-dn.net/?f=m%3D%5Crho_%7Bwater%7D%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%29)
Since the diameter of the raindrop is 3 millimeters, the radius is 1.5 mm or 0.0015 meters. Substituting we get:
![m=(0.98\times10^3\frac{kg}{m^3})(\frac{4}{3}\pi(0.0015m)^3)](https://tex.z-dn.net/?f=m%3D%280.98%5Ctimes10%5E3%5Cfrac%7Bkg%7D%7Bm%5E3%7D%29%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%280.0015m%29%5E3%29)
Solving the operations:
![m=1.39\times10^{-5}kg](https://tex.z-dn.net/?f=m%3D1.39%5Ctimes10%5E%7B-5%7Dkg)
Now, we substitute the values in the formula for the drag force:
![F_d=(1.39\times10^{-5}kg)(9.8\frac{m}{s^2})](https://tex.z-dn.net/?f=F_d%3D%281.39%5Ctimes10%5E%7B-5%7Dkg%29%289.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%29)
Solving the operations:
![F_d=1.36\times10^{-4}N](https://tex.z-dn.net/?f=F_d%3D1.36%5Ctimes10%5E%7B-4%7DN)
Now, we substitute in the formula:
![1.36\times10^{-4}N=\frac{1}{2}C\rho_{air}Av^2](https://tex.z-dn.net/?f=1.36%5Ctimes10%5E%7B-4%7DN%3D%5Cfrac%7B1%7D%7B2%7DC%5Crho_%7Bair%7DAv%5E2)
Now, we solve for the velocity:
![\frac{1.36\times10^{-4}N}{\frac{1}{2}C\rho_{air}A}=v^2](https://tex.z-dn.net/?f=%5Cfrac%7B1.36%5Ctimes10%5E%7B-4%7DN%7D%7B%5Cfrac%7B1%7D%7B2%7DC%5Crho_%7Bair%7DA%7D%3Dv%5E2)
Now, we substitute the values. We will use the area of a circle:
![\frac{1.36\times10^{-4}N}{\frac{1}{2}(0.45)(1.21\frac{kg}{m^3})(\pi r^2)}=v^2](https://tex.z-dn.net/?f=%5Cfrac%7B1.36%5Ctimes10%5E%7B-4%7DN%7D%7B%5Cfrac%7B1%7D%7B2%7D%280.45%29%281.21%5Cfrac%7Bkg%7D%7Bm%5E3%7D%29%28%5Cpi%20r%5E2%29%7D%3Dv%5E2)
Substituting the radius:
![\frac{1.36\cdot10^{-4}N}{\frac{1}{2}(0.45)(1.21\frac{kg}{m^{3}})(\pi(0.0015m)^2)}=v^2](https://tex.z-dn.net/?f=%5Cfrac%7B1.36%5Ccdot10%5E%7B-4%7DN%7D%7B%5Cfrac%7B1%7D%7B2%7D%280.45%29%281.21%5Cfrac%7Bkg%7D%7Bm%5E%7B3%7D%7D%29%28%5Cpi%280.0015m%29%5E2%29%7D%3Dv%5E2)
Solving the operations:
![70.67\frac{m^2}{s^2}=v^2](https://tex.z-dn.net/?f=70.67%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%3Dv%5E2)
Now, we take the square root to both sides:
![\begin{gathered} \sqrt{70.67\frac{m^2}{s^2}}=v \\ \\ 8.4\frac{m}{s}=v \\ \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Csqrt%7B70.67%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%7D%3Dv%20%5C%5C%20%20%5C%5C%208.4%5Cfrac%7Bm%7D%7Bs%7D%3Dv%20%5C%5C%20%20%5Cend%7Bgathered%7D)
Therefore, the velocity is 8.4 m/s