# Maximum Height Problem

Students were given the following problem:

A dense stone is tossed vertically into the air with an initial speed of 25 m/s. Represent how ball’s motion changes in a “step-by-step” diagram. By elaborating and specifying aspects of your diagram, determine the maximum height.

Students had some brief direct instruction on what it means to do make “step-by-step” diagram and then to elaborate the diagram to solve a problem. The work below represent the first time students are practicing the approach, so keep that in mind. I’ve included a good example from a student of what this method might look like when done well, followed by two others that aren’t quite able to work it out.

**Good Example**

**Student Approach #1**

**Student Approach #2**

#1 How do you think students #1 and Students #2 were thinking about the problem and/or the their approach to the method they were asked to practice? In what way does their work “make sense” to them?

#2 What questions would you like to ask student #1 and student #2 to probe into their thinking?

#3 What questions would you like to ask student #1 and student #2 to*nudge* their thinking along?

This is my first attempt to post on “Physics Mistakes”, so pity me if I forget decorum…

#1

Both focused adequately on how the velocity would decrease over time. Neither accounted for (perhaps by displaying double arrows for acceleration that changes velocity or by giving examples of how the velocities were calculated) why the velocity changes, but it probably made sense in their heads. Their work differs in that Student #1 attended to the average velocity, and Student #2 added time information. Each thought that they could “add” the velocities in appropriate ways to get the displacements. Whereas Student #1 added the average velocities on each time interval displayed, Student #2 added the instantaneous velocities (or perhaps just the initial velocities) at the endpoints of each interval. Student #2’s writing of the velocity in the middle of the interval might indicate forgetting that the written velocities were instantaneous. Both students neglected to consider that the intervals did not all have the same time interval. I like the idea of including heterogeneous time intervals to probe understanding. Did any students attempt to make the diagram more Motion Map-like by showing the changes every 0.5 seconds? Were they more successful at calculating displacement?

#2

First, “How did you calculate the displacement?”, to anchor the conversation.

Then, ask about the first interval. “How long did it take? How far did it go? How do you know?”

Then, move on to the other intervals or perhaps jump to the last one. “How long did it take to slow from 5 m/s to 0 m/s?”

#2

“When and where was the stone/ball traveling up at 15 m/s?”

“I noticed that you didn’t draw the arrow all the way to the maximum height in the last time interval. Can you tell me why?”

I’m sure there are more questions to ask, but that’s all that comes immediately to mind. For both students, once they have demonstrated their understandings, one might ask, “It seems like a lot of work to calculate the displacement by adding up all the smaller displacements. Is there an easier way to calculate the displacement here?”