### An Analysis of the Monkey and Hunter Problem Using the Graphing Calculator

The TI-83 can be used to recreate one of the classic questions of physics: What should a monkey in a tree do when the gun is pointed straight at him -- stay where he is? Or, drop down? Using the graphing calculator, students easily see that the bullet has two types of motion (horizontal motion and vertical motion) that are totally independent of one another. They determine which parameters need to be changed in order to affect the horizontal motion of the bullet. Also, they determine that the bullet and the monkey fall at the same rate.

Assume that the monkey in the tree is located 3 m above the ground. Since we are graphing motion, assume that the point on the ground directly beneath the gun is the origin. The gun is located a horizontal distance of 10 m away from the monkey in the tree. The gun is held at the same level as the monkey, at a height of 3 m.

Assume the monkey drops as soon as he sees the smoke from the gun when it is fired. Have students guess how long it will take the monkey to fall 3 m. Students should predict the accelerated motion of the monkey as it falls.

Students should try different values for the horizontal speed of the bullet coming out of the gun. Ask student questions such as: What happens if the horizontal speed of the bullet is increased or decreased? What determines how long it takes the bullet to reach the ground? What determines how far the bullet travels horizontally? What happens if the horizontal separation is increased? Decreased? What would be the effect of launching the bullet at an angle?

By observing the motion of the monkey and of the bullet, students discover that the horizontal motion of the bullet is independent of its vertical motion. They will observe that the bullet moves the same distance horizontally with each unit of time. Students also discover that the bullet and the monkey fall at the same rate.

### TI - 83 Solution to the Monkey and Hunter Problem:

Mode settings:

Parametric mode
Simultaneous equations

Equations: Using the parametric setting on the graphing calculator allows one to enter equations that describe the vertical and the horizontal motion of the monkey and of the bullet. x1T and x2T will represent the horizontal motion of the monkey and of the bullet, respectively. y1T and y2T will represent the vertical motion of the monkey and of the bullet, respectively. Enter these parametric equations under y=. Our equations are of the form:

d(t) = do + vit + 1/2at2
• Monkey equations:
• x1T = 10 + 0 t + 0.5 (0) t2
Since the monkey is a horizontal distance of 10 m away from the gun, its
do=10. It has no horizontal velocity or acceleration, so those terms are zero.
• y1T = 3 + 0 t + 0.5 (-9.8) t2
Since the monkey is located 3 m above the ground, its do=3. When it drops, its
initial vertical velocity is 0 m/s, down, so vi=0. The acceleration due to gravity
causes it to accelerate downward at -9.8 m/s2.
• Gun equations
• x2T = 0 + v t + 0.5 (0) t2
Since the gun is located at the origin, do=0. Students will choose different values
for the horizontal speed of the bullet, v. There is no horizontal acceleration, so
a=0.
• y2T = 3 + 0 t + 0.5 (-9.8) t2
Since the gun is located at the origin, do=0. When the bullet is fired, it begins to
drop, so its initial vertical velocity is 0 m/s, down, or vi=0. The acceleration due
to gravity causes it to accelerate downward at -9.8 m/s2.

Window settings

Tmin=0
Tmax= (have students guess how long it will take monkey to fall 3 m)
Tstep=0.01
xmin=0
xmax=12
xscale=10
ymin=-2
ymax=4
yscale=3

Hints:

• Students should enter different values for v, the horizontal speed of the bullet. This is done by using the STO function of the calculator. Example: students choose a horizontal speed of 10 m/s. Students will enter
10 STO V
• You can use the animate setting on the calculator to change the "monkey" into a 0.
• You can freeze the screen by pressing Enter.
• The 2nd draw 1 command can be used as a replay.
• If you use dot mode on the mode settings, the plotting is quicker