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A Terminal Situation
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by David Castro
Charles A. Dana Center
University of Texas
Austin, Texas
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Problem Statement
In order to address the needs of both AP Physics B and Physics C students, the first mechanics question from the 1975 AP Physics C Exam (reproduced below) is solved twice in this exercise -- once conceptually and once using formal calculus notation.
When an object is dropped within the atmosphere, it experiences a resistive force that depends on the object's speed. This causes significant deviations from the ideal free-fall conditions usually assumed in AP Physics C problems. Despite these complications, AP Physics C students who master the mathematical skills required to understand this type of motion will be well equipped to solve a related set of physics problems, including RC and LR circuits, in addition to applications of Faraday's law. These applications, as well as an overview of previously released AP Physics C terminal velocity problems, are discussed in the solution and commentary below. Although a formal study of this phenomenon requires the use of differential calculus, AP Physics B students can gain a strong conceptual understanding of the real-world behavior of falling objects through reflection and graphical analysis.
1975 Mech 1
A sphere of mass m is released from rest. As it falls, the air exerts a retarding force on the sphere that is proportional to the sphere's velocity (FR = -kv). Neglect the buoyancy force of the air.
(A) On the circles below, draw vectors representing the forces acting on the sphere: - just after it is released, and
- after it has been falling for a long time and reached terminal velocity.
Give each vector a descriptive label.
(B) Determine the terminal velocity of the sphere.
(C) Draw the following three graphs for the sphere's motion, clearly showing significant features of the motion just after the sphere is released, as well as after a long time. - Acceleration as a function of time
- Velocity as a function of time
- Position as a function of time
Click here to view the answers and commentary!
David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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