Question

3) The velocity v(t) of a skydiver falling to the ground is governed by the equation m dv/dt mg-kv, where g is the acceleration due to gravity, and k>0 is the drag constant associated with air resistance a) Find the analytical solution for V(t), assuming v(0) 0 b) Find the limit of v(t) as t goes to infinity. This is known as the terminal velocity. c) Give a graphical analysis of this problem, and re-derive the formula for the terminal velocity.

Answer 1

(a) First obtain the analytical solution for v(t) assuming v(0) = 0

dv .2 = mg - kv- dt
  
dv m = df mg k - v2

by dividing by k and separating the variables

dv k dt mg m

k tC arctanh mg m Tmg
through integrating table using $\small a^2=\frac{mg}{k}$

solving for v:

gm -tanh gk C (t) m

applying boundary condition v(0) = 0 tells us C = 0

gm v(t) = gk tanh k m

b) By finding the limit of v(t) as t approaches infinity leads to terminal velocity

gm lim [v(t) = (1)
as  
lim tanh(t)1

so the terminal velocity is

qm k

c)  A graphical analysis of this problem, and re-derive the formula for the terminal velocity

V' (ft/s2) 33 2 gm V k V (ft/s) 4 3 1 2 -1 4 -2 -3 -1 gm -2