**Goal:** Relate friction, velocity, and time.

**Source:** UMPERG

A cart rolls across a table two meters in length. Half of the length of

the table is covered with felt which slows the cart at a constant rate.

Where should the felt be placed so that the cart crosses the table in

the least amount of time?

- On the first half of the table
- Centered on the table
- On the second half of the table
- It doesn’t matter where the felt is placed
- None of the above
- Cannot be determined

## Commentary:

## Answer

The

felt should be placed on the second half of the table. After the cart

rolls across the felt it will travel at a lower speed. To minimize the

time to cross the table one must minimize the time the cart spends at

the lower speed. The graph to the right illustrates the point for the

two extreme cases: felt on first half (gray curve) and felt on second

half (black curve). The velocity vs. time graph for the case where the

felt is on the second half of the table is above the velocity vs. time

graph for all other possibilities. Answer (3) is the best choice.

## Background

Students should have some experience using the concepts of velocity and

acceleration to solve kinematics problems and analyze graphs. The

question students need to answer is what configuration will permit the

cart to travel at a higher speed for the longest period of time (or the

lowest speed for the shortest period of time). A graph provides support

for a conceptual argument.

Issues to consider: (1) Can students reason and analyze a situation

involving constant acceleration. (2) Do students try to solve the

problem using algebraic methods? (2) Can students use graphical methods

and conceptual reasoning? (3) Can students verbalize the central idea —

an object will travel a certain distance in less time if its speed is

higher?

## Questions to Reveal Student Thinking

Where is the cart moving the fastest? … the slowest?

What does a graph of the velocity vs. time look like?

How do you determine when the cart has reached the end of the table from

a graph of velocity vs. time?

## Suggestions

Try some limiting cases. If the piece of felt were small (say 10 cm)

but slowed the cart from 1 to .8 m/s on a 3m table. Approximately how

long would the trip take if the felt were placed at the beginning of the

table?…at the end of the table?