For the free$-$falling parachutist $($i$.$e$.,$ the jumper$)$ problem discussed in the lecture, we

assumed the linear air drag that the air resistance force is described as $F_U=-cv,$ and develop the

ODE model for parachutist velocity as

$\frac{dv}{dt}=g-\frac{c}{m}v$

Now rather than the linear relationship for $F_U,$ you might choose to model the upward force on the

parachutist as a second$-$order relationship as $F_U=-c^{\text{'}}v|v|,$ where $c^{\text{'}}=$ a bulk second$-$order drag

coefficient $(k\frac{g}{m}).$ Note that the second$-$order term could be represented as $v^2$ if the parachutist

always fell in the downward direction. For the present case, we use the more general

representation, $v|v|,$ so that the proper sign is obtained for both the downward and the upward

directions. With the new assumption to the air force, you may develop the ODE model for the

velocity field as

$\frac{dv}{dt}=g-\frac{c^{\text{'}}}{m}{v}^2$

$($a$)$ Using calculus, obtain the closed$-$form solution $($i$.$e$.,$ analytical solution$)$ for the case where

the jumper is initially at rest at $t=0.[10$ pts$]$

$($b$)$ Using Euler's method, perform the numerical calculation of the velocity till $t=12s$ with a

time step size of $2$ s $.$ Use the same initial condition and parameter values

$68\mathrm{.}1$ kg $),$ but with second$-$order drag coefficient as a value of $0\mathrm{.}225k\frac{g}{m}$ for $c^{\text{'}}.$ Compare the

analytical solution and numerical solutions by plotting the results a same figure. $[10$ pts$]$