Computer Algebra Recipes

4 INTRODUCTION
(a) At what angle Á with the horizontal is the ball thrown?
(b) How long does it take the ball to reach the fence?
(c) Plot the entire trajectory, then animate the motion of the ball, including
the fence in the animation.
Let's choose the origin to be on the ground below the initial position of the
ball and take the x-coordinate to be horizontal and the y-coordinate vertical.
To begin the recipe, we ¯rst clear Maple's internal memory of any previously
assigned values (other worksheets may be open with numerical values given to
some of the same symbols being used in the present recipe). This is done by
typing in the restart command after the opening prompt ( >) symbol, ending
the command with a colon (:), and pressing Enter (which generates a new
prompt symbol) on the computer keyboard.
> restart:
All Maple command lines must be ended with either a colon, which suppresses
any output, or a semicolon (;), which allows the output to be viewed.
Next, the given parameter values are speci¯ed. For example, the initial
x-coordinate of the ball is entered, the symbolic name xb being placed to the
left of the Maple assignment operator (:=). The numerical value (0) of the
coordinate is placed on the right-hand side of the operator and the output
suppressed here with a command-ending colon. Assigned quantities can be
mathematically manipulated. In a similar manner, the numerical values of the
ball's initial y-coordinate (yb), the horizontal location (xf) of the fence, the
fence's height (yf), the initial speed (V) of the ball, and the magnitude of the
gravitational acceleration (g) are entered. Because the command entries are
short, we have chosen to place them all on the same prompt line, separating
the entries by a space for reading clarity.
> xb:=0: yb:=2: xf:=20: yf:=3.5: V:=15: g:=9.8:
Using the symbol * for multiplication, we express the horizontal (vx) and vertical
(vy) components of the ball's initial velocity in terms of the unknown angle
Á. The Maple input syntax phi is used to generate the Greek letter Á in the
output. Note that the assigned value (15) of V is automatically substituted.
> vx:=V*cos(phi); vy:=V*sin(phi);
vx := 15 cos(Á) vy := 15 sin(Á)
Using the standard kinematic relations [Oha85], we calculate the ball's x and y
coordinates at arbitrary time t. The symbols +, -, /, and^ are used for addition,
subtraction, division, and exponentiation. Note that the decimal coe±cient of
t 2 in the output is given to 10 digits, Maple's usual default accuracy.
> x:=xb+vx*t; y:=yb+vy*t-(1/2)*g*t^2;
x := 15 cos(Á) t y := 2 + 15 sin(Á) t ¡ 4:900000000 t 2
Setting x=xf in the solve command, the time t = tf for the ball to reach the
fence is determined in terms of Á.
> tf:=solve(x=xf,t); #time to reach fence