A2L Item 011
- Description: Assess the validity of a solution procedure for airplane aborting takeoff.
- Goal: Analyze and evaluate a solution to a given problem.
- Source: UMPERG
- Keywords: Acceleration, Mechanics, Problem Solving
The question for students:
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?
- 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 for teachers:
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.