2. A Conceptual Approach To Teaching Kinematics Mark Rosenquist and Lillian McDermott A Review by Marsha Turin and Lisa Henderson TechBoston Academy

3. Topics in Kinematics Instantaneous velocity as a limit Distinguishing between position and velocity Distinguishing velocity, change of velocity and acceleration Making connections among graphs, concepts and motions

4. Instantaneous Velocity As A Limit The concept was developed via lab experimentation and pencil and paper exercises involving nonuniform motion. The concept was defined based on the students lab and pencil and paper observations in order to make the concept concrete. The concept was defined as the time interval that motion appeared uniform.

5. Summary of Methods Lab Exercise Examination of ticker tape motion diagrams generated by an accelerating fan cart and a ticker tape timer. Examine the dots for uniform motion. Pencil and Paper Exercise Students plotted a magnified section of a curved position vs. time graph until the motion appears uniform.

6. Distinguishing Between Position and Velocity Misunderstanding Students incorrectly interpret position vs. time graphs. Same speed Same speed x t

7. Addressing The Misunderstanding Used lab demonstrations to concretely distinguishing position and velocity over time. Part I 2 balls, 2 tracks, one ball moves uniformly, the other accelerates 2 balls, 2 tracks, balls move uniformly, one ball starts ahead of the other 2 balls, 2 tracks,balls move with different uniform velocities, one ball passes the other.

8. Addressing The Misunderstanding Part II 2 balls, 2 tracks, one ball has positive acceleration, the other has a negative acceleration, no passing takes place. v t x t

9. Distinguishing Velocity, Change of Velocity and Acceleration Clarification of these concepts as distinct Helps students associate a particular velocity with a particular time Helps students recognize the necessity of considering the time interval during which a change in velocity occurs Requires students to apply the definition of acceleration in comparing two real motions

10. Instantaneous Velocity Experiment Students are asked to compare the motion of a ball rolling down a track with that of a marked revolving belt of constant velocity 2-3 meters Belt marked in 10 cm segments

11. Comparing Belt and Ball Speed Students determine belt speed by timing movement of a given mark over a given distance As the ball rolls down the incline its velocity, relative to the constant velocity belt changes

12. Students Gain Immediacy of Trial and Results This differs from traditional methods as the velocity is measured directly rather than as a derivation of position graphs Velocity at any moment is therefore determined during the experiment rather than some time after t v

13. Timing the ball’s Progress Students start a stopwatch when they release the ball When the ball’s velocity matches that of the belt the stopwatch is stopped Students measure the time it takes for the ball to attain this velocity Students plot data for several speeds and trials

14. Combining Immediacy and Tactile Experience Students see instantaneous velocity as they see the ball’s motion overtake that of the constant speed belt The physical act of starting and stopping the stopwatch draws student’s attention to the changing time intervals They see the ball speed up and have to respond to how quickly the increase in speed occurs

15. Increased Complexity with 2 Moving Balls This is followed by comparing the motion of two balls rolling down inclines of differing degree Students analyze the accelerations of the two balls by comparing their relative positions and motion

16. Comparing the Motion of Two Moving Balls A B Greater initial velocity, then greater acceleration Separation constant

17. Discussion and Analysis Citing the moment that the ball’s motion is the same, students incorrectly assess that acceleration is the same. Instructor is then able to direct students to “instantaneous” velocities Students are able to separate the concepts of velocity and acceleration through analysis of their own observations

18. Graphs Students graph the motions they have seen Create situations that will match graphs Recognize that different graphs can represent the same motion using different axes

19. ACTIVE PHYSICS LOOKS AT ACCELERATION Students are asked to analyze the motion with respect to the spacing of the dots Students recognize that the spacing of the dots correlates to the motion that they observed in the moving cart * * * * * * * * * * * * * *

20. STUDENTS LOOK AT SEGMENTS OF THE MOTION Students are asked to look at the segment where the motion is fastest and slowest They correlate what they “know” with what they observe in the dots This brings them close to an understanding of instantaneous velocity

21. Carl Lewis and Graphs Students are asked to determine how accurately they can assess Carl Lewis’s speed. Students recognize that they are given 10 meter increments and cannot immediately determine an “instantaneous” velocity, although this term is not introduced in the text, the opportunity for the teacher is there

22. Two Motions As the time it takes to cover 10 m decreases… Velocity increases