# Day 56: A Graph to Visualize Average Velocity

We started Constant Acceleration in college-prep today. Rather than dive right in with carts and motion detectors, I propped up one end of a lab table with textbooks (best use ever) and let a C-battery roll down. (*Batteries accelerate more slowly than marbles and hot wheels cars. They also roll much straighter.)*

“What do you observe?” I asked

“It rolls down and gets faster.” they said.

“Prove it. You have 10 minutes.” I challenged them. I hate prescribing directions for activites like this. I want to see how my students approach these tasks.

They wanted stopwatches and metersticks. Some wanted tape.

One group wisely rolled the battery down a whiteboard and left marks at one second intervals. (They were done in 2 minutes.) Most marked out equal intervals of distance to time.

Most groups made data tables to show that it takes less time to travel each successive distance interval, thereby showing it continously increases in speed. Many groups added a velocity column and calculted the “velocity” for each interval to show it changes. (*But velocity when? where? average? *I didn’t want to go down that road just yet. I just let it be.)

Some groups went further and also made distance-time graphs of their data to show the slope increases.

Two groups went even further and added an average-velocity step graph like the one above.

**It was beautiful.** And something I had never considered doing.

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You see, over the years, I’ve tried a variety of acceleration labs. Kids would collect position-time data and make position-time and velocity-time graphs. And getting the velocity-time graph was always laborious. Here are some methods I’ve tried in years past…

Method 1: Manually draw tangent lines on the position-time graphs. Calculate and graph the slopes of the tangent lines. (Tedious)

Method 2: Use the slope tool in Logger Pro to get the slope of the tangent at each data point. Graph the slopes of the tangent lines. (Computer issues)

Method 3: Kids calculate the average velocity for each distance/time interval. Tell them to graph it at the midpoint in time. This typically involved a lot of hand waving b/c kids didn’t quite understand why at the midtime rather than the end time. And I’d still have groups that would incorrectly graph the average velocity at the end time. One time I made a data table worksheet to avoid this issue — but it was scary table with rows in between rows for midtime data.

Method 4: Method 3, but using Excel (*OMG, what was I thinking?*)

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**The average velocity step grap method is perfect.** It doesn’t matter how the students took the data. They calculate the average velocity for each interval, then graph each average velocity as a step that is as long as the interval. No need to handwave about midtimes. No need to assume the acceleration is constant.

The board pictured above inspired me, so I had all groups make their own average velocity step graph as well, just to see if it would work.

“Is this how the velocity-time graph really looks?” I asked.

“No. There wouldn’t be any steps. It would be a line. Or a curve.” they said.

They made the leap *on their own* to draw a line *through* the steps. And, lo and behold, the “best fit line” cuts through the middle of each step — the midtime.

**You can’t miss it. A great visualization.**

Kids who took data at even time intervals had equal sized step-widths and step-heights. Kids who took data at even distance intervals had unqual step-widths and step-heights (the steps got narrower and shorter over time — which in a data table looks like non-constant acceleration). *But the line still cut through the midtime of each step.* Now we can talk about why that happened and what that means AFTER, rather than all the handwaving and number crunching first.

Several graphs also got a y-intercept, which we chalked up to reaction time error.

**I love it when I learn from kids!**

Frank, I like this intro a lot. I too struggled with helping the kids generate the v vs. t graph. I now get the v vs t graph in a very straight-forward way: roll a low friction car down a ramp (from rest) and have it go through two photogates…set up the photogates to collect two time intervals, time interval for complete path and time interval for cart-flag (bolt) blocking bottom photogate. The prelab discussion is very rich and the students immediately realize that any velocity measurement is really an average velocity measurement. The discussion always turns toward how instantaneous velocities are measured. Students are quick to realize that it is impossible to measure an instantaneous velocity. The only way to "get" an inst. velocity is to make the time interval shorter and shorter. Students realize that the time interval for the flag blocking the photogate is extremely small and realize that they can actually use this velocity measurement as an inst. velocity. They move the bottom photogate further away from the top photogate as they conduct the experiment…so they end up plotting an x vs. t graph (predictably curvy) and a v(inst) vs. t graph (linear with an intercept). The basics of the model is easily teased out from a post-lab discussion.Rob