Topic 2: Position-Time Graphs

Here, one-dimensional motion is considered and rt = d is introduced. The difference between average speed and instantaneous speed is emphasized. This is practiced in different situations. The justification for this is that to understand the hang time problem, one-dimensional speed must be understood first.
Standards:

1. 8PC1.b. Students know that average speed is the total distance traveled divided by the total time elapsed and that the speed of an object along the path traveled can vary.
2. 8PC1.c. Students know how to solve problems involving distance, time, and average speed.
3. 8PC1.f. Students know how to interpret graphs of position versus time (and graphs of speed versus time) for motion in a single direction.

Featured software

• Ultrasonic motion with output to CCgraph. Students experiment with walking a graph. See jnlp: Simple Motion Challenge
• Moving Man with output to CCgraph. Physical motion or a sketched graph can control motion of the man.
• Problem poser. This is vaporware that will be able to ask many questions requiring d = rt problems in different contexts and with different values.

Classroom discussion. Emphasize that speed is a ratio. Units. Also, d/t is an average that is different than the speed at any one time. Distinguish between average and instantaneous values. For instance, suppose that there were two ways to get somewhere: freeway all the way (where you travel at v1) or take a shortcut where you have to travel at v2 < v1 but the distance is shorter by x. Which one to choose? Race.

Investigations. Use the timer to measure the elapsed time for someone running, things falling, etc.
Extensions. Use distance-time and velocity-time graphs to control models. Two objects, one moving at constant speed the other at different speeds. Show graphs of the motion of both objects in both distance-time and velocity-time mode. Challenge: manipulate the graphs, keeping the constant velocity object constant, so that they arrive at their destination at the same time (in other words, their average velocities are the same). What can you say about the relationship between the two graphs when this goal has been achieved?
Suggested lab. Use a photogate timer to measure speeds of different objects. Perhaps we can use a camera as a photogate timer.
Assessments. Lots of rt = d (and r = d/t, t = d/r) problems. Be sure to include atomic and astronomical scales as well as the speeds of familiar objects.