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}}]