**Goal:** Interrelate representations of acceleration and identifying potential misconceptions.

**Source:** UMPERG

Which of the following statements are true?

A. While an object moves at constant speed, its acceleration must be zero.

B. For a ball to roll up a hill and then back down, its acceleration must change.

C. When an object’s velocity versus time graph crosses the time axis, its acceleration must be zero.

- Only (A)
- Only (B)
- Only (C)
- Both (A) and (B)
- Both (A) and (C)
- Both (B) and (C)
- All three are true
- None is true
- Cannot be determined

## Commentary:

## Answer

Answer (8) is the best choice. An object’s acceleration is nonzero if its direction of motion is changing, or its speed is changing, or both. A constant speed does not imply zero acceleration because the object’s direction of motion could be changing. Therefore (A) is false. An object will have a nonzero acceleration if its velocity is changing, even if its velocity is (instantaneously) zero. Therefore (C) is false. For a ball that rolls up a hill and then back down, the acceleration can be constant. This will be the case if the hill has a constant slope, and the friction and air resistance forces are small. Therefore (B) is also false.

## Background

Assessment Issues: (1) Do students know that an object has a nonzero acceleration whenever its speed or direction of motion change? Do they think that the speed must change for there to be a nonzero acceleration? (2) Do students confuse velocity and acceleration? Do they think that the acceleration is positive/zero/negative whenever the velocity is positive/zero/negative? (3) Do students use graphs and pictures to answer this question?

## Questions to Reveal Student Reasoning

## Suggestions

Have one group of students perform some motion (perhaps by walking or moving an object), and challenge another group to graph position versus time, velocity versus time, or acceleration versus time for that motion. (If the motion performed is in two dimensions, students should graph one of the components of the motion.)