91Ó°ÊÓ

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

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),
$$\begin{gathered} m=\frac{1}{2}a \\ ⇒a=2m \end{gathered}$$

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