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).

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

20

Recorder

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.