Measuring Position-Time and Velocity-Time

One must get the methods of measuring position and time working together if we are to obtain a reliable representation of motion. To that end we have developed a unique system composed of elements gathered at garage sales, kitchen, laundry, washroom, toy store and garage.

A timing helmet (Radio Shack Road Patrol, slightly modified) which can be worn during an excursion flashes a red light and beeps obnoxiously once every second. (That's 86400 times a day. We counted!) Two bingo markers are attached to the ends of a metre stick and we unroll a strip of paper towelling along side the route. As I ride, every time my helmet beeps, I stab the paper towel with bingo marker, thus producing a spot showing where I was every second along the way.

The result of one passage at low speed looks something like this.

I've numbered the points and sound effects with a verbal annotation. Using the back edge of the spot to indicate the positions we can make a table of position and time.

Time (s)	Position (m)
0	−5.0
1	−4.7
2	−4.0
3	−3.0
4	−2.0
5	−1.1
6	0.0
7	1.0
8	2.0
9	3.1
10	4.1
11	4.7
12	4.9

A graph of these data looks like this.

The blue circles are the points we measured and the light-blue line is the probable position vs. time function which varies smoothly as it passes through the points.

After speeding up during the first two seconds, the cart travels at a fairly constant velocity for the next 8 seconds. During these 9 seconds it travels 8 m. So the average velocity is 1 m/s. Because the line is straight during this time, the instantaneous velocity is also 1 m/s there.

When the line is curved, the average and instantaneous velocities are most likely not the same. The average velocity from 0 s to 2 s is the slope of the light green line which passes through the points at 0 s and 2 s. Its slope is 0.5 m/s. The instantaneous velocity at 1 s is the slope of the dark green line. This is the line which is tangent to the velocity curve at the 1 s mark. Its slope is visibly less than 0.5 m/s. By measuring the slopes of the tangent lines to the curve at every second during the trip (and more often if you wish) one can tabulate the instantaneous velocities at those times and then make a velocity-time graph of the trip. Some aspects of linear motion are more easily seen on velocity-time graphs than on position-time graphs, and some are more visible on position-time graph than on velocity-time graphs. Therefore, it is useful to be able to translate between the two of them.

Here is the velocity-time graph of the trip. Compare the two graphs and convince yourself that they represent the same motion.