A body of mass m moves in the +x direction, subject only to drag force: Fx= (-1/2)(CD*rhoair*S*vx^2)

Question

With drag coefficient CD, air density rho air, cross sectional area S, and velocity component vx. At time t=0 the body is at postion x=0 and has velocity of vx=v0 where v0 is some positive value.
a. Calculate the velocity of the body as a function of time, vx(t) for t >=0:
b. Does the body come to rest in a finite time? Remember you must justify your answer.
c. Calculate the position of the body as a function of time, x(t) for t>=0
d. Does the body come to rest in a finite distance, i.e. cover only a finite distance? Remember you must justify your answer.

Answer 1

m dv/dt = -[(1/2) rho Cd S] v^2
dv/dt = - [(1/2) rho Cd S/m] v^2
let k = [(1/2) rho Cd S/m]
then
dv/dt = -k v^2

dv/v^2 = -k dt

-1/v = -kt - c
v = 1/(kt+c)
when t = 0, v = Vo
Vo = 1/c
so c = 1/Vo
so
v = 1/(kt +1/Vo)
when does v = 0?
only when t = oo
dx/dt = 1/(kt+r) where r = 1/Vo
dx = dt/(kt+r)
x = (1/k)ln(kt+r)+c2
when t = 0, x = 0
0 = (1/k) ln r + c2
c2 = -(1/k) ln r
so
x = (1/k)(ln(kt+r)-ln r)
x = (1/k) ln [ (kt+r)/r) ]
remember r = 1/Vo
x = (1/k) ln [Vo kt + 1} that does not become constant
check details, I did that fast but perhaps you get the idea