Question

As an astronaut visiting Planet X, you’re assigned to measure the free-fall acceleration. Getting out your meter stick and stopwatch, you time the fall of a heavy ball from several heights. Your data are as follows:

Analyze these data to determine the free-fall acceleration on Planet X. Your analysis method should involve fitting a straight line to an appropriate graph.

Short answer

The free-fall acceleration of Planet X is 8 m/s².

Step-by-step solution

Step 1. Write the given information.

The table of several heights with respective falling times is given.
Determine the velocity of the ball with respect to each case

Step 2. Determine the velocity of each case

To determine the free-fall acceleration of Planet X, plot the graph between the height (m) and the time²(s²). The graph plotted between these two parameters gives the best-fitted graph. The slope obtained from the graph gives the values of free-fall acceleration.
Following is the table which is to be plotted on the graph

Height (m)	Fall time (s)	time2(s2)
0.0	0.00	0.00
1.0	0.54	0.29
2.0	0.72	0.52
3.0	0.91	0.82
4.0	1.01	1.02
5.0	1.17	1.36

Now, plot the graph between the Height (m) and Fall time² (s²)

This is the best-fitted graph to obtain the straight line.
The equation of motion of the straight line is given by
$h=m{t}^2+C....\left(1\right)$
Here, C is the constant and it is equal to zero as the graph has no intercept and m is the slope of the straight line.
Now using the equation of motion for the free-fall object is

$h=\frac{1}{2}a{t}^2.....\left(2\right)$
Comparing the equation (1) and (2),
$$\begin{gathered} m=\frac{1}{2}a \\ \Rightarrow a=2m \end{gathered}$$

Now, determine the slope of the straight line,

localid="1648556493542" $$\begin{gathered} m=\frac{(h_4-h_2)}{({t_4}^2-{t_2}^2)}=\frac{(4-2)m}{\left(1.02-0.52\right)s^2} \\ m=\frac{2}{0.5}=4m/s^2 \end{gathered}$$
Therefore, the free-fall acceleration will be
$$\begin{gathered} a=2m=2\left(4\right) \\ a=8m/s^2 \end{gathered}$$
The free-fall acceleration of Planet X is 8 m/s²