Acceleration day two
After the numbers generated on day one did not well support a numerical approach to d=½at² I opted to start the second day by returning to last week and the relationship that for a constant velocity the equation is d = vt and on a time versus distance graph there is a constant slope, a constant velocity. The initial velocity v₀ is the same as the final velocity. Transferred to a time versus velocity graph the data produces a slope equals zero horizontal line.
On the board above the material to the left of the dotted line is a recap of the previous week. Then I recapped what I was trying to do on Monday: start from a speed v₀ equals zero and then go faster and faster in a linearly increasing rate of increased velocity. On Monday I left as unknown what shape might be produced on a time versus distance graph if the velocity increased at a steady rate versus time. On Monday the data produced a half of a parabola.
What I could not know on Monday was whether my velocity had increased linearly as intended.
Spreadsheet work after class showed that I had increased my velocity linearly. I printed the above chart and another chart out for class on Wednesday. This allowed me to argue that for the above graph there must be a y = mx+ b relationship, specifically v = at + v where v₀ was arranged to be zero meters per second by starting from a speed of zero.
I noted that on Monday a parabola had been generated and that the formula I used, d = mt² appeared to model the data points.
I then noted that one can sometimes see places that one cannot or will not get to go from a mountaintop. I told the class I would show them something from a mountaintop in mathematics called calculus, a system that would give us the equations for the accelerating system.
As I wrote each equation I noted how choices that I made eliminated some of the terms. v₀ was set to be zero by my stationary start on Monday, and I started from a distance d₀ of zero as well. I briefly noted that y=ax²+bx+c has a slope and a y-intercept, something that is often omitted in college algebra courses but that I feel the student should see. I also noted that the ½ appears as a result of integration, I did not try to further explain the ½ that appears in the distance equation.
I then circled back to a promise I had made on Monday that I would show the class a full parabola with the RipStik. The diagram above on the lower left was notes on the procedure I would follow. At this point it was perhaps 12:24 PM.
The sidewalk was premarked at 0, 3, 6, and 9 meters. After a first run I added a 10 meter mark. I left the turn-around, up around 11 meters, unmarked. The vertex works out better if not timed, at least based on past experience.
Three timers were deployed.
Two timers produced usable data, and their data sets closely concurred. The parabolic model did slice through the data points suggesting the possibility that a parabolic model was sufficient to explain the data.
This approach felt better than the geometric edge of calculus approach I attempted fall 2019. This exercise leaves the class well primed for measuring the acceleration of gravity and tracing the arc of a ball on Friday.
On the board above the material to the left of the dotted line is a recap of the previous week. Then I recapped what I was trying to do on Monday: start from a speed v₀ equals zero and then go faster and faster in a linearly increasing rate of increased velocity. On Monday I left as unknown what shape might be produced on a time versus distance graph if the velocity increased at a steady rate versus time. On Monday the data produced a half of a parabola.
What I could not know on Monday was whether my velocity had increased linearly as intended.
I noted that on Monday a parabola had been generated and that the formula I used, d = mt² appeared to model the data points.
I then noted that one can sometimes see places that one cannot or will not get to go from a mountaintop. I told the class I would show them something from a mountaintop in mathematics called calculus, a system that would give us the equations for the accelerating system.
As I wrote each equation I noted how choices that I made eliminated some of the terms. v₀ was set to be zero by my stationary start on Monday, and I started from a distance d₀ of zero as well. I briefly noted that y=ax²+bx+c has a slope and a y-intercept, something that is often omitted in college algebra courses but that I feel the student should see. I also noted that the ½ appears as a result of integration, I did not try to further explain the ½ that appears in the distance equation.
I then circled back to a promise I had made on Monday that I would show the class a full parabola with the RipStik. The diagram above on the lower left was notes on the procedure I would follow. At this point it was perhaps 12:24 PM.
The sidewalk was premarked at 0, 3, 6, and 9 meters. After a first run I added a 10 meter mark. I left the turn-around, up around 11 meters, unmarked. The vertex works out better if not timed, at least based on past experience.
Three timers were deployed.
Two timers produced usable data, and their data sets closely concurred. The parabolic model did slice through the data points suggesting the possibility that a parabolic model was sufficient to explain the data.
This approach felt better than the geometric edge of calculus approach I attempted fall 2019. This exercise leaves the class well primed for measuring the acceleration of gravity and tracing the arc of a ball on Friday.
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