**Goal:** Analyze and evaluate a solution to a given problem.

**Source:** UMPERG

#### In order to solve the problem:

An airplane accelerates down a runway in order to take off

but aborts and applies brakes causing the plane to stop. The plane

speeds up at a constant rate for 5 seconds, then slows down at the same

rate when the brakes are applied. The plane stops at a point that is

100 meters from its initial position. What was the acceleration of the

airplane during the first 5 seconds?

#### Someone suggests the following procedure:

(A) The acceleration of the plane is constant and the same for the

entire motion.

(B) The entire process takes 10 seconds and the displacement is 100

meters.

(C) It is possible, therefore, to use the formula “change in x =

v_{o,x} t + 1/2 a_{x} t^{2}“, where

v_{o,x} is zero and t = 10s.

(D) The only unknown in this equation is a_{x}, so solve for it.

Which of the following is true?

- The procedure is invalid because statement A is incorrect.
- The procedure is invalid because statement B is incorrect.
- The procedure is invalid because statement C is incorrect.
- The procedure is invalid because statement D is incorrect.
- The procedure is invalid because more than one statement is incorrect.
- The procedure is valid.

## Commentary:

## Answer

(5) More than one statement is incorrect. The

acceleration is not constant for the entire motion and so (A) is

incorrect. Although the magnitude of the acceleration is constant its

direction changes. Statement (C) is incorrect because the formula in

(C) is only valid over periods the acceleration is constant.

## Background

This question requires students to make decisions and judgements which

are needed when solving kinematics problems with understanding. The

kinematics equations are of limited use. They apply directly only to

problems involving constant acceleration. Students are usually not

aware of this limitation and are apt to apply the kinematics expressions

much too broadly. Students also tend to view acceleration as a scalar

quantity and therefore see the acceleration as constant even when it is

not so.

## Questions to Reveal Student Thinking

How do we determine the acceleration. What is the acceleration while

the plane is speeding up? … slowing down? If necessary ask the

following. What is the direction of acceleration while the plane is

speeding up? … slowing down?

## Suggestions

Draw a graph of velocity vs. time for constant acceleration. Draw a

graph of velocity vs. time for the problem situation. Discuss the

acceleration and displacement in terms of these graphs.