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ETTING
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29
Activity 5—Rolling ball
notes for teachers
Concepts
Function explored: parabolic.
Plotting a ball rolling down a ramp of varying
inclines creates a family of curves, which can be
modeled by a series of quadratic equations. This
activity investigates the values of the coefficients in
the quadratic equation,
y
=
ax
2
+
bx
+
c
.
Materials
Ÿ
calculator
Ÿ
CBR
Ÿ
calculator-to-CBR cable
Ÿ
mounting clamp
Ÿ
large (9 inch) playground ball
Ÿ
long ramp (at least 2 meters or 6 feet—a
lightweight board works well)
Ÿ
protractor to measure angles
Ÿ
books to prop up ramp
Ÿ
TI ViewScreen (optional)
Hints
Discuss how to measure the angle of the ramp. Let
students get creative here. They might use a
trigonometric calculation, folded paper, or a
protractor.
See pages 6–12 for hints on effective data
collection.
Typical plots
15
¡
30
¡
Typical answers
1. the third plot
2. time; seconds; distance of object from CBR; feet
or meters
3. varies (should be half of a parabola, concave
up)
4. a parabola (quadratic)
5. varies
6. varies (should be parabolic with increasing
curvature)
7. 0
¡
is flat (ball can’t roll); 90
¡
is the same as a
free-falling (dropping) ball
Explorations
The motion of a body acted upon only by gravity is
a popular topic in a study of physical sciences. Such
motion is typically expressed by a particular form of
the quadratic equation,
s
= ½
at
2
+
v
i
t
+
s
i
where
0
s
is the position of an object at time
t
0
a
is its acceleration
0
v
i
is its initial velocity
0
s
i
is its initial position
In the quadratic equation
y
=
ax
2
+
bx
+
c
,
y
represents the distance from the
CBR
to the ball
at time
x
if the ball’s initial position was
c
, initial
velocity was
b
, and acceleration is 2
a
.
Advanced explorations:
Since the ball is at rest when released,
b
should
approach zero for each trial.
c
should approach the
initial distance, 0.5 meters (1.5 feet).
a
increases as
the angle of inclination increases.
If students model the equation
y
=
ax
2
+
bx
+
c
manually, you may need to provide hints for the
values of
b
and
c
. You may also direct them to
perform a quadratic regression on lists
L1
,
L2
using
their calculators. The ball’s acceleration is due to
the earth’s gravity. So the more the ramp points
down (the greater the angle of inclination), the
greater the value of
a
. Maximum
a
occurs for
q
= 90
¡
, minimum for
q
= 0
¡
. In fact,
a
is
proportional to the sine of
q
.