velocity vs. clock time from position vs. clock time data

Velocity vs. Clock Time from Position vs. Clock Time Data

The first column of the figure below shows the clock times at which a ball accelerating down a uniform incline from rest passes each of a series of positions, with positions separated by 15 cm.

The second column shows the time interval required for each 15-cm distance. The fourth column shows the average velocity of for each 15-cm distance.
The third column shows the midpoint clock time of each interval.

clock time	`dt	mid time	vAve
0	1.6	0.8	9.375
1.6	1.1	2.15	13.63636
2.7	1	3.2	15
3.7	0.8	4.1	18.75
4.5	0.6	4.8	25
5.1

Below we see the corresponding graph of velocity vs. clock time, where the fourth column is plotted vs. the third..

The graph appears to be reasonably linear. It is therefore plausible that the acceleration of the object is uniform.

To determine the acceleration we find the average rate at which velocity changes.

This average rate is equal to the change in velocity divided by the change in clock time, along the best-fit straight line.

The change in velocity, from the left most to the right most point of the straight line, is from about 6 cm/second to 27 cm/s. This is a change of approximately 21 cm/second.

This change occurs between clock times 0 and six seconds. The change thus requires 6 seconds.

The average rate of velocity change is therefore approximately 21 cm / sec / [ 6 sec ] = 3.5 cm / sec / sec.

The equation of the best-fit line has slope 3.5407, very close to the above estimate.