^{1}and Lillian C. McDermott

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### Abstract

In this paper we describe a long-term, large-scale investigation of the ability of university students to treat velocity and acceleration as vectors in one and two dimensions. Some serious conceptual and reasoning difficulties identified among introductory students also were common among pre-college teachers and physics graduate students. Insights gained from this research guided the development of instructional materials that help improve student learning at the introductory level and beyond. The results have strong implications for the teaching of undergraduate physics, the professional development of teachers, and the preparation of teaching assistants.

The authors gratefully acknowledge the collaboration of many members of the Physics Education Group. Costas Constantinou, Gregory F. Francis, Mark D. Somers, and Stamatis Vokos participated in the development of curriculum. Contributions to the research were made by Bradley S. Ambrose, Sean M. Courtney, Paula R. L. Heron, MacKenzie R. Stetzer, and John R. Thompson. Special thanks are due to the instructors whose classes were included in this investigation. We also deeply appreciate support from the Division of Undergraduate Education and the Division of Physics of the National Science Foundation.

I. INTRODUCTION

II. MOTIVATION FOR THE EMPHASIS ON VECTORS AND OPERATIONAL DEFINITIONS

III. OVERVIEW OF THE INVESTIGATION

IV. PROBLEMS USED FOR DISTINGUISHING VECTOR SKILLS FROM CONCEPTUAL UNDERSTANDING

A. Test on 1D problem: Colliding carts

B. Tests on 1D matched problems: Concept application and vector manipulation

C. Test on 2D problem: Vector manipulation

D. Commentary

V. PROBLEMS USED FOR PROBING CONCEPTUAL UNDERSTANDING

A. 1D pretest: Ball moving up and down an inclined ramp

B. 2D pretest: Object moving along a closed, horizontal track

VI. ANALYSIS OF INCORRECT RESPONSES ON 1D AND 2D PRETESTS

A. Incorrect reasoning about kinematics at arbitrary points along a trajectory

1. Not recognizing that instantaneous velocity is tangent to the trajectory

2. Not distinguishing between velocity and acceleration and sometimes using identical vectors for both

3. Mistakenly assuming that the acceleration is zero because the speed is constant

4. Mistakenly assuming that the acceleration is directed toward special points

B. Incorrect reasoning for a turnaround point

1. Mistakenly using a nonzero vector for the velocity at the turnaround point

2. Mistakenly assuming that the acceleration is zero at a turnaround point

C. Incorrect reasoning for the point at which an object starts from rest

1. Not treating the instantaneous velocity as zero for an object starting from rest

2. Mistakenly assuming that the instantaneous acceleration is zero for an object starting from rest

3. Mistakenly assuming that the instantaneous acceleration has a radial component for an object starting from rest

D. Incorrect (or incomplete) reasoning about application of dynamics to kinematics

1. Not associating the direction of the acceleration with that of the net force

2. Confusing net force and acceleration

E. Commentary on results from 1D and 2D pretests

VII. DEVELOPMENT OF TUTORIAL CURRICULUM

A. Kinematics in one dimension

B. Kinematics in two dimensions

VIII. ASSESSMENT OF 1D AND 2D TUTORIALS

A. 1D post-tests

1. Collision of two pucks on frictionless table

2. Motion of two blocks up and down an inclined ramp

3. Commentary

B. 2D post-tests

1. Motion with changing speed along a closed horizontal trajectory

2. Pendulum

3. Commentary

IX. APPLICATION TO PRECOLLEGE TEACHER PROFESSIONAL DEVELOPMENT

X. CONCLUSION

### Key Topics

- Kinematics
- 118.0
- Graduates
- 15.0
- Physics education
- 9.0
- Lectures
- 8.0
- Undergraduates
- 5.0

## Figures

Questions used early in the investigation. In each case, students were asked to draw velocity vectors and to indicate the approximate direction of the acceleration for various points in the motion.

Questions used early in the investigation. In each case, students were asked to draw velocity vectors and to indicate the approximate direction of the acceleration for various points in the motion.

Questions used to examine the relation between vector skills and conceptual understanding: (a) 1D pretest on collision of two carts, (b) 1D matched pretests on manipulation of vectors, and (c) 2D post-test on manipulation of vectors.

Questions used to examine the relation between vector skills and conceptual understanding: (a) 1D pretest on collision of two carts, (b) 1D matched pretests on manipulation of vectors, and (c) 2D post-test on manipulation of vectors.

Examples of pretests administered to large numbers of students. Students were asked to draw velocity and acceleration vectors at various points during each motion. (a) 1D pretest on ball moving up and down a ramp. (b) 2D pretest on object moving at constant speed along a closed, horizontal track. Some students were also asked about the case that the object speeds up from rest.

Examples of pretests administered to large numbers of students. Students were asked to draw velocity and acceleration vectors at various points during each motion. (a) 1D pretest on ball moving up and down a ramp. (b) 2D pretest on object moving at constant speed along a closed, horizontal track. Some students were also asked about the case that the object speeds up from rest.

Examples of 1D post-tests. (a) Two pucks collide on a frictionless table. Students were asked to give the directions of the accelerations and the relative magnitudes. (b) Two blocks of different mass move on a frictionless incline. Students were asked to indicate the direction of the acceleration of block B at the earlier instant and to draw the change in velocity vector for block A.

Examples of 1D post-tests. (a) Two pucks collide on a frictionless table. Students were asked to give the directions of the accelerations and the relative magnitudes. (b) Two blocks of different mass move on a frictionless incline. Students were asked to indicate the direction of the acceleration of block B at the earlier instant and to draw the change in velocity vector for block A.

Example of 2D post-test. Students were asked to sketch velocity and acceleration vectors at the labeled points for the case that the object moves at constant speed and to draw acceleration vectors for the case that the object moves with increasing speed.

Example of 2D post-test. Students were asked to sketch velocity and acceleration vectors at the labeled points for the case that the object moves at constant speed and to draw acceleration vectors for the case that the object moves with increasing speed.

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