# Differential Equation direction fields

• Sep 15th 2008, 05:54 PM
njr008
Differential Equation direction fields
The equation for a stone falling under gravity with air resistance,
dv/dt = g - kv^2, v > 0,
where k is a constant drag coefficient. Pay particular attention to the direction along the line v = sqrt{g/k} = vt. What is the significance of vt?
Sketch solution graphs which start at t = 0,
(i) for an initial velocity > vt and
(ii) for an initial velocity < vt.
I know about putting in values to draw the direction fields, but this question has thrown many people in our class and our lecturer said it was easy, so any help would be greatly appreciated.
• Sep 15th 2008, 07:04 PM
skeeter
$v_t = \sqrt{\frac{g}{k}}$ is terminal velocity ... note that $a = \frac{dv}{dt} = 0$ when $v = v_t$.

for $v < v_t$ , the speed will increase at a decreasing rate, with $v \to v_t$ as $t \to \infty$.

for $v > v_t$ , the speed will decrease at a decreasing rate, with $v \to v_t$ as $t \to \infty$.

hope that helps with your sketch.
• Sep 16th 2008, 06:01 AM
shawsend
Quote:

Originally Posted by njr008
I know about putting in values to draw the direction fields, but this question has thrown many people in our class and our lecturer said it was easy, so any help would be greatly appreciated.

Hey, if I shoot a bullet straight up, how fast is it going when it hits the ground? Not too right? Same way if I start up there. If I shoot it down faster than terminal velocity, it will slow down, if I shoot it slower, it will speed up. So the plots look like the one I drew below: asymptotic to terminal velocity right? Here's the Mathematica code if you want to experiment with it:

Code:

line = Graphics[{Dashed,     Line[{{0, N[Sqrt[16/0.05]]},       {5, N[Sqrt[16/0.05]]}}]}]; slow = NDSolve[{Derivative[1][v][t] ==       g - k*v[t]^2 /. {g -> 16, k -> 0.05},     v[0] == 1}, v, {t, 0, 5}] slowplot = Plot[v[t] /. Flatten[slow],   {t, 0, 5}] fast = NDSolve[{Derivative[1][v][t] ==       g - k*v[t]^2 /. {g -> 16, k -> 0.05},     v[0] == 40}, v, {t, 0, 5}] fastplot = Plot[v[t] /. Flatten[fast],   {t, 0, 5}, PlotRange -> All] Show[{slowplot, fastplot, line},   PlotRange -> {{0, 5}, {0, 50}}]