M.2.1 Instantaneous Velocity, Average Velocity, and Acceleration

Below is a sample set of demonstrations with an air track for illustrating these concepts using a clock (the "white clock") that measures the elapsed time for a glider to travel one meter and another clock (the "red clock") that measures the time for the 0.1m flag of the glider to pass a sensor at the end of the one meter interval.

1. First explain exactly what the clocks are measuring to the students. A transparency is available that can be projected during the demonstrations to remind them what is measuring what. Show them that the white clock measures the elapsed time for the glider to travel one meter by passing your hand through the start-gate, counting off "one thousand one, one thousand two, —" for several seconds, and then passing your hand through the stop-gate. Then show them that the red clock measures the time for the glider flag to pass its sensor by blocking its gate with the glider flag, counting off several seconds, and removing the glider.

Then if the white clock reading is labeled T and the red clock reading is labeled t:

average velocity = 1 meter/T

instantaneous velocity = 0.1 meter/t

At some point you may wish to discuss how the exact instantaneous velocity is defined in terms of the calculus derivative by imagining the flag length to become smaller and smaller.

2. To check that everyone understands what the clocks are measuring, ask them, "If the track is level and I send a glider through the gates, what will be the relationship between the readings of the two clocks? (and then do the demo!) Answer: red clock reading = 1/10 white clock reading.

3. Now use a block to tilt the track up. Ask, "The red clock should now read (greater than, less than, the same as) 1/10 the white clock. In other words, is the instantaneous velocity at the end of one meter of acceleration (greater, less, the same) as the average velocity over the one meter distance?"

4. Now use a larger mass glider. "The clock readings should be (greater, less, the same as) before?"

5. "At what fraction of the one meter distance does the glider attain an instantaneous velocity equal to the average velocity over the one meter? In other words, where should the red sensor be placed to get red = 1/10 white with the track tilted?" (Answer: 1/4 meter)

6. You can check the measured acceleration against the tilt of the track. If the track length is L and it is tilted height h,

acceleration = a = g sin q = gh/L

Then from the white clock, a = 2 meters/t2 a = 2 meters/T 2

and from the red clock, a = (0.1m/t)2/2m = 0.005 meters/t2