Computer Algebra Recipes

120 CHAPTER 3. LINEAR ODE MODELS
> phi:=Phi[1]: #select angle
In this case, the time tf to reach the fence is calculated
> tf:=evalf(tf);
tf := 1:715218642
to be about 1.72 s, slightly more than the 1.68 s without air resistance.
Setting y = 0, we determine the time T for the ball to hit the ground,
> T:=solve(y=0,t);
T := 2:090175405; ¡0:1946627301
the positive angle (¯rst solution here) being selected.
> T2:=T[1]; #choose positive time
T2 := 2:090175405
The ball hits the ground after 2.09 seconds, compared to 2.06 seconds without
air resistance.
The fence is plotted as a quite thick (default color red) line.
> fence:=plot([[xf,0],[xf,yf]],style=line,thickness=3):
Using exactly the same syntax as in the introductory recipe, the motion of the
ball is animated with the fence as background.
> animate(pointplot,[[[x,y]],symbol=circle,symbolsize=14],
t=0..T2,frames=200,background=fence,scaling=constrained);
To see this animation, execute the recipe on your computer, click on the opening
frame,andthenonthestartarrowintheMapletoolbar.
PROBLEMS:
Problem 3-7: Maximum drag coe±cient
With all other parameters the same as in the text recipe, what is the maximum
value of the drag coe±cient k such that the ball just clears the fence? What
are the two corresponding initial angles?
Problem 3-8: How high?
If the drag coe±cient k is equal to 0:1s ¡1 , how high can the fence be for the
ball to just clear it, all other parameters the same as in the text recipe? What
are the two initial angles in this case?
Problem 3-9: A falling raindrop
For a sphere of diameter d meters moving in air, the approximate value of the
constant k in Stokes's linear resistance law, ~ Fdrag = ¡k~v newtons, is given by
k =1:55 £ 10 ¡4 d. For a small spherical raindrop (density ½ =103kg/m3 ,
d =10 ¡4 m) falling from rest, determine the distance through which it falls in
t seconds and its velocity then. Plot the distance and velocity separately over
atimeintervalt = 0 to the time at which the velocity reaches 99 percent of its
terminal (asymptotic) value.