# A2L Item 011

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 =
vo,x t + 1/2 ax t2“, where
vo,x is zero and t = 10s.

(D) The only unknown in this equation is ax, so solve for it.

Which of the following is true?

1. The procedure is invalid because statement A is incorrect.
2. The procedure is invalid because statement B is incorrect.
3. The procedure is invalid because statement C is incorrect.
4. The procedure is invalid because statement D is incorrect.
5. The procedure is invalid because more than one statement is incorrect.
6. The procedure is valid.

### Commentary:

(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.