Question: Falling RaindropsAssume that when a rain drop begins to fall, both its radius and velocity are zero.Then as the rain drop falls, its radius increases linearly with respect to time. That is,r=kt where k is constant. The mass of the rain drop ism=Kr3where the radius r is the function of t as assumed above and K is another constant.(a) Write the formula

Falling Raindrops

Assume that when a rain drop begins to fall, both its radius and velocity are zero.

Then as the rain drop falls, its radius increases linearly with respect to time. That is$,$

$r=kt$ where $k$ is constant. The mass of the rain drop is

$m=K{r}^3$

where the radius $r$ is the function of $t$ as assumed above and $K$ is another constant.

$($a$)$ Write the formula for $m(t),$ the mass of the raindrop as a function of time.

$($b$)$ Newton's second law of motion states

$\frac{dp}{dt}={\displaystyle \sum _{?}^{?}}$forces acting $on$ the body

where $p=mv$ is the momentum. Here we assume the only force is gravity, giving

${\displaystyle \sum _{?}^{?}}$ forces $=mg.$ Therefore, we have the following initial value problem:

$\frac{dp}{dt}=mg$;$,v(0)=0$

Using the product rule, show that this initial value problem is equivalent to:

$t{v}^{\text{'}}+3v=tg$;$,v(0)=0.$

$($c$)$ Using the method of solving first order linear ODEs to solve the initial value problem

for $v(t).$