**Goal:** Associate velocity graph with physical motion.

**Source:** UMPERG

A soccer ball rolls across the road and down a hill as shown below.

Which of the following sketches of v_{x} vs. t represents the

horizontal velocity of the ball as a function of time?

## Commentary:

## Answer

(5) None of the above. The ball crosses the road in a straight line at

a more-or-less constant speed (perhaps slowing down slightly) provided

that the road is in good condition and the rolling friction between the

ball and road is sufficiently small. As the ball rolls down the hill it

will speed up, and so there will be an acceleration in the direction of

motion, with a non zero component to the right. The following graph is

a reasonable representation of the horizontal velocity as a function of

time.

## Background

This problem could challenge students in several areas: (1) Can

students recognize how the velocity is changing? What criteria do they

use? (2) Do students realize that as the ball moves down the hill it

speeds up and the x-component of velocity increases? Students may

associate the increase in velocity with the y-direction only. (3) Do

students associate the graph with the terrain over which an object

travels? The process of translation of a motion quantity to a graph can

be very difficult for students. (4) Will students confuse motion

quantities? When students analyze the graphs of velocity vs. time they

may be interpreting the graphs in terms of position instead of velocity.

## Questions to Reveal Student Reasoning

Is the velocity ever zero? Where does the ball speed up? …slow down?

What is the direction of the velocity while the ball is on the sloped

section? Does the velocity have a non-zero horizontal component?

## Suggestions

Set up a demonstration with a horizontal surface and a ramp, both with

the same net horizontal displacement. Roll a ball slowly across the

horizontal surface and down the ramp. Ask students to judge which

horizontal displacement took more time. Over what section (horizontal

surface or ramp) is the velocity larger on average?