Chapter 4, Problem 2P

A Falling Ball In a physics experiment a lead ball is dropped from a height of 5 m. The students record the distance the ball has fallen every one-tenth of a second. (This can be done by using a camera and a strobe light.) Their data are shown in the margin.

1. (a) Make a scatter plot of the data.
2. (b) Use a calculator to find a power model.
3. (c) Use your model to predict how far a dropped ball would fall in 3 s.

Time (s)	Distance (m)
0.1	0.048
0.2	0.197
0.3	0.441
0.4	0.882
0.5	1.227
0.6	1.765
0.7	2.401
0.8	3.136
0.9	3.969
1.0	4.902

To determine

To sketch: The scatter plot of the data.

Explanation of Solution

Given:

In a physics experiment a lead ball is drops from a height of 5 m, the students’ record the distance the ball has fallen every one-tenth of a second listed below in Table 1.

$\mathrm{Time}\left(\mathrm{s}\right)$	$\mathrm{Distance}\left(m\right)$
0.1	0.048
0.2	0.197
0.3	0.441
0.4	0.882
0.5	1.227
0.6	1.765
0.7	2.401
0.8	3.136
0.9	3.969
1.0	4.902

Table 1

Calculation:

Use online graphing calculator and draw the scatter plot of the given data as shown below in Figure 1.

Form Figure 1, it is noticed that the scatter plot is an increasing function.

To determine

To find: The power model for the given data by using calculator.

Answer to Problem 2P

The power model that best fits the data point for the given data is $s\left(t\right)=4.96221226\times t^{2.00268961}$, where $s$ be the distance travelled by the ball in $t$ seconds.

Explanation of Solution

Let $s\left(t\right)$ be the distance travelled by the ball in $t$ seconds.

Use online graphing calculator and obtain the power regression line as follows.

Input time data as x values and $s\left(t\right)$ as y value.

From above screenshot, it is noticed that the power model that best fits the data point is $s\left(t\right)=4.96221226\times t^{2.00268961}$.

Therefore, the power model that best fits the data point for the given data is $s\left(t\right)=4.96221226\times t^{2.00268961}$, where $s$ is the distance travelled by the ball in $t$ seconds.

To determine

To predict: The distance for a dropped ball would fall in 3 seconds.

Answer to Problem 2P

The fallen distance of the dropped ball in 3 seconds $s=44.7920m$.

Explanation of Solution

From part (b) the power model that best fits the data point for the given data is $s\left(t\right)=4.96221226\times t^{2.00268961}$, where $s$ is the distance travelled by the ball in $t$ seconds.

Substitute $t=3$ in the power model and calculate the distance fall by the dropped ball in 3 second as follows,

$\begin{array}{c}s\left(t\right)=4.96221226\times 3^{2.00268961}\\ =4.96221226\times 9.026633\\ =44.7920\mathrm{m}\end{array}$

Therefore, the fallen distance of the dropped ball in 3 seconds is $s=44.7920m$.