Arc of a Sphere

02 September 2008

In this activity we will explore the shape of the path through the air made by a sphere. The path is a path in two-dimensional space. The graph will be a space versus space graph using the MKS system.

We will be working outside to measure the arc of the sphere. The set-up and variables to be measured are seen in the following diagram.

arc of a ball

Where k is the height of the y-intercept above the x-axis
and r is the distance from the axis of symmetry to one x-intercept (root).

Task	n	Position title	Student name	x	y
Right x-intercept	1	Ball underhand pitcher
	2	Meter stick holder
	3	Recorder
right mid arc height	4	Tape measure holder		x	y
	5	Tape base holder
	6	arc height observer
	7	Data recorder
Vertex height k	8	Tape measure holder	Same as 4	x	y
	9	Tape base holder	Same as 5
	10	Vertex position observer
	11	Data recorder
left mid arc height	12	Tape measure holder		x	y
	13	Tape base holder
	14	arc height observer
	15	Data recorder
Left x-intercept	16	Ball catcher		x	y
	17	Meter stick holder
	18	Recorder
x-intercept to x-intercept	19	Wheel roller		distance = 2r
x-intercept to x-intercept	20	Recorder

Theory

$y = - (\frac{k}{r^{2}}) x^{2} + k$

Does our data agree with the theory? Use a spread sheet to plot the data. Set up a table like the seen below for spring 2008. Make an xy scattergraph of all three columns. Use the k and r from the activity to calculate the predicted path. The function below is an example based on the spring 2008 data. Your values of k and r will be different. Spring 2008 k was 2.2 m and r was 3.0 meters. The presumes that the column titles are in row 1 and that the first x-value is in cell A2. This formula would be in C2 and can be "filled down" for the next four rows.

=-(2.2/(3.0^2))*A2^2+2.2

Data from spring 2008

Data for the arc of a sphere
location x (m)	y1 actual height (m)	y2 predicted height using the equation(m)
-3.0	0.0	0.00
-1.5	1.3	1.65
0.0	2.2	2.20
1.5	1.4	1.65
3.0	0.0	0.00