Activity:
To determine and examine the path taken by a bottle cap projectile on the
earth and on the moon.
Mathematical
Skills: grade
level 912, graphing
and solving quadratic equations, graph interpretation, using a graphing
utility (a TI84+ graphing calculator was used here)
Materials:
graph paper, graphing calculator
Directions:
Earth: 
On Earth, it is possible to shoot a bottle cap 64 feet straight up into
the air with a rubber band. In
t
seconds after firing, the bottle cap is
s(t)
= 64t  16t^{2}
feet above your hand.
a.) 
Draw a graph that simulates
the position of the bottle cap on earth. 
b.) 
How long does it take the
bottle cap to reach its maximum height? 
c.) 
What is the maximum height
reached by the bottle cap? 



Moon: 
On the moon, the same force will send
the bottle cap to a height of
s(t) = 64t  2.6t^{2}
feet in t seconds.
a.) 
Draw a graph that simulates
the position of the bottle cap on the moon. 
b.) 
Approximately, how long does
it take the bottle cap to reach its maximum
height? 
c.) 
Estimate the maximum height
reached by the bottle cap to the nearest tenth of
a foot? 

For the teacher:
While this problem can be
solved with paper and pencil, the use of the graphing calculator will
offer students a visual interpretation of the problem and reinforce
graphing concepts.
One possible solution using
the TI84+ graphing calculator:
Type the equation
into Y1.
Discuss why using the variable "x" will yield the
same result as using the variable "t".


Determine a window by trial and error or
by a prior algebraic solution obtained by setting the function
equal to zero and solving for t.

Hit graph. Is this value seen
in the window after hitting
"trace"
the actual maximum value?


Find the actual maximum value using the
calculator's maximum function.
(found under "calc" above the "trace"
key)

The maximum function will ask for
a left bound. Move the "spider"
(cursor) to the left of where you
imagine the maximum to be located.
Hit enter.


Move the "spider" to the right
of where you imagine the maximum to be located.
Hit enter.

When asked to "guess", just
press "enter." Notice the pointers
surrounding the area where the
maximum will occur.


Hitting "enter" will yield
the
x and y values of the maximum.

Repeat the process for
the
second equation.


Possible window. Students should predict if the time span will increase and discuss why or why not.

Answer.


Students should discuss
why the same force created different paths for the bottle cap on
the earth and on the moon. This is a good opportunity to engage
in "writing in the mathematics classroom." 
