Finding Accelerations from Average Velocity Data

The table below shows ave. velocity vs. clock time data for positions 15 cm apart (see also Velocity vs. Clock Time from Position vs. Clock Time Data). Each average velocity is associated with the midpoint clock time of the corresponding 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

To find the approximate accelerations we must determine `dv / `dt for each pair of average velocities.

From the first to the second average velocity the change in velocity is approximately 4.4 cm/second, and the interval between the midpoint times is approximately 1.35 seconds, so that the acceleration is approximately 3.3 cm / sec^2.

Accelerations for the remaining 3 intervals between the midpoint times are calculated similarly and are indicated in the table.

We note that the original data were taken from an actual experiment submitted by a student.

The data were good for the equipment used, and the velocity vs. clock time graph shown below is reasonably linear.

However, due to unavoidable errors in observation, the point-to-point slopes vary considerably, even though it is entirely plausible that the acceleration is uniform.

In addition to showing how accelerations can be inferred from position vs. clock time data, this analysis shows how small errors in observation can magnify themselves in a two-step process such as this one, where position vs. clock time data are used to approximate velocity vs. clock time, and the resulting velocity vs. clock time is used to predict the acceleration vs. clock clock time.

mid time	vAve	`dv	`dt	a = `dv / `dt
0.8	9.375	4.4	1.35	3.3
2.15	13.63636	1.4	.95	1.5
3.2	15	3.8	.9	4.2
4.1	18.75	6.3	.7	9
4.8	25

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