# Arc of a Sphere

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.

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).

Right x-intercept1Ball underhand pitcher
2Meter stick holder
3Recorder
right mid arc height4Tape measure holderxy
5Tape base holder
6arc height observer
7Data recorder
Vertex height k8Tape measure holderSame as 4xy
9Tape base holderSame as 5
10Vertex position observer
11Data recorder
left mid arc height12Tape measure holderxy
13Tape base holder
14arc height observer
15Data recorder
Left x-intercept16Ball catcherxy
17Meter stick holder
18Recorder
x-intercept to x-intercept19Wheel rollerdistance = 2r
20Recorder

## Theory

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

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)
-3.00.00.00
-1.51.31.65
0.0 2.22.20
1.5 1.41.65
3.0 0.00.00