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A2L Item 009

Meta-Information
Description: 
Evaluate the effects of friction early in your trip, versus late in your trip, on the trip time.
Goal: 
Relate friction, velocity, and time.
Level: 
Undergraduate
Keywords: 
Acceleration
Keywords: 
Displacement
Keywords: 
Friction
Keywords: 
Mechanics
Keywords: 
Reasoning
Keywords: 
Velocity
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?

  1. On the first half of the table
  2. Centered on the table
  3. On the second half of the table
  4. It doesn’t matter where the felt is placed
  5. None of the above
  6. 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?