A ball is thrown vertically and data are collected at various times in its flight.


Time (sec)

0.5

1.0

1.5

2.0

2.5

3.0

Distance (m)

7.3

12.1

14.6

14.4

11.8

7.0

Assume that air resistance can be ignored, then the height of the ball satisfies the quadratic equation:

h(t) = v0t - (g/2)t2,

due to gravity. (Note: There is no constant term as we are assuming that the height of the ball is zero at t = 0.)

a. Use the Excel's trendline to find the best constants v0 and g that fit the data in the table and write that equation in your report. (Remember that when you are using trendline, you must decide if your graph passes through the origin. Does this one?) Graph both the quadratic function and the data. Find the time that your model predicts the ball will hit the ground. Also, find how high the ball goes, and find the time that it reaches this highest point.

b. The average velocity between two times t1 and t2 is given by the formula:

vave = (h(t2) - h(t1))/(t2 - t1).

Create tables showing the average velocity of the ball based on the t1 and t2 values given below.


t1

1

1

1

1

1

1

t2

2

1.5

1.2

1.1

1.05

1.01

t1

2

2

2

2

2

2

t2

3

2.5

2.2

2.1

2.05

2.01

t1

2.5

2.9

2.95

2.99

2.999

t2

3

3

3

3

3

As seen in the lecture notes, the velocity of the ball at a given time is the derivative of the height function at that time. Compute the derivative of h(t),

h'(t) = v(t).

Evaluate v(1), v(2), and v(3). How do these values compare to the values of vave that you obtained for each of the tables above?

c. Associate vave with t1, i.e., let vave = v(t1). Once again compute vave for the table below, then graph v(t) versus t using these data. Describe the graph that you have produced. Use trendline (or any other method) to find the equation of this graph and find the v and t-intercepts. Compare this equation to the equation of the derivative h'(t) that you obtained above.

t1

0

0.5

1

1.5

2

2.5

3

t2

0.01

0.51

1.01

1.51

2.01

2.51

3.01