Computer Algebra Recipes

INTRODUCTORY RECIPE 5
tf := 4 1
3 cos(Á)
A comment, pre¯xed by the pound sign #, has been added to the command
line. Short comments are useful for later reference or for others to read and
understand the purpose of a Maple command. 1
A transcendental equation eq for Á results on evaluating y at t = tf and
equating the result to yf:
> eq:=eval(y,t=tf)=yf;
20 sin(Á) 8:711111111
eq := 2 + ¡
cos(Á) cos(Á) 2 =3:5
The transcendental equation is solved for Á, the result being labeled ©.
> Phi:=solve(eq,phi); #angles in radians
©:=0:9917653601; 0:6538908144; ¡2:149827293; ¡2:487701839
Four angles, expressed in radians, are generated in the above output ©, the
default accuracy being 10 digits. Since the initial angle must be above the horizontal,
only the positive answers are acceptable as solutions for this problem.
Since there are two positive results, this means that Daniel could throw the ball
at two di®erent angles to just clear the fence. To proceed, we shall select one of
the positive answers, say the second one in ©. This is done by entering Phi[2].
You can look at the ¯rst positive angle by changing this entry to Phi[1]. Ifdesired,
the angle Á ¼ 0:65 radians can be converted to degrees using the convert
command with units as the second argument.
> phi:=Phi[2]; theta:=convert(phi,units,radian,degree);
Á := 0:6538908144 μ := 37:46518392
In this case, Daniel throws the ball at an angle μ ¼ 37 1 degrees to the horizontal.
2
It should be mentioned that the convert command is very useful for converting
an expression from one form to another, the form of conversion being dictated
by the choice of second argument. To see the types of conversions possible with
Maple, click on the convert command in the worksheet, then on Help in the
tool bar, and ¯nally on Helponconvert.
Using the °oating-point evaluation command, evalf, we numerically evaluate
the time to reach the fence, which is found to be about 1.68 seconds.
> tf:=evalf(tf); #time to reach fence
tf := 1:679846933
To plot the entire trajectory, the time T for the ball to hit the ground must
be determined. This is accomplished by setting y = 0 and solving for t, which
produces two answers.
> T:=solve(y=0,t);
T := ¡0:1981184874; 2:060197767
1 Longer or more detailed comments may be inserted into a worksheet by clicking on
Insert in the tool bar at the top of the computer screen, then on Execution Group, on
either Before Cursor or After Cursor, onText, and ¯nally typing in the comments.