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|  | Introduction
In AP Physics C, students often have difficulty using calculus appropriately and intuitively. This issue can be addressed by helping students see parallels in the way that calculus is applied in a wide variety of physics contexts. There are few better opportunities to make these comparisons than by comparing the motion of a falling body as it approaches terminal velocity to RC and RL circuits as they approach steady-state conditions. Both of these situations involve creating a differential equation based on a fundamental physical principle and solving the equation to find an important physical quality as a function of time. Furthermore, the differential equation, once solved, can be further differentiated or integrated to find additional meaningful physical information.
Consider the major mathematical techniques required to perform the three following tasks.
| Step |
Process |
Motion Problem |
RC Circuit Problem |
LR Circuit Problem |
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Problem statement |
An object is dropped from rest. As it falls, it experiences a friction force of FR = -kv. Find an expression for velocity, acceleration, and displacement of the object as a function of time. |
A circuit consists of a source of EMF, a switch, a capacitor, and a resistor. At t = 0 sec, the switch is closed. Find an expression for the charge on the capacitor, the circuit current, and the energy stored in the capacitor. |
A circuit consists of a source of EMF, a switch, an inductor, and a resistor. At t = 0 sec, the switch is closed. Find an expression for the current in the inductor and the energy stored in the inductor. |
| 1 |
Fundamental law or principle |
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| 2 |
Mathematical expression of the physical situation |
mg - kv = ma |
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| 3 |
Express changing quantity as a differential |
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Already a differential |
| 4 |
Separation of variables |
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| 5 |
Solve differential equation |
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| 6 |
Differentiation |
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| 7 |
Integration |
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This problem is based on the second electricity and magnetism problem on the 1998 AP Physics C Exam. Note that the original problem has been modified to encourage students to use as much calculus as possible.
It is strongly recommended that you and your students work through the February 2005 AP Physics Featured Question: A Terminal Situation. This problem includes a detailed discussion of the calculus that is used in the solution.
Problem Statement
In the circuit shown above, the switch S is initially in the open position shown, and the capacitor is uncharged. A voltmeter (not shown) is used to measure the correct potential difference across resistor R1.
(A) On the circuit diagram above, draw the voltmeter with the proper connections for correctly measuring the potential difference across resistor R1.
(B) At time t = 0, the switch is moved to position A. Determine the voltmeter reading for the time immediately after t = 0.
(C) At some later time, t, a measurement of potential difference across R1 is again taken. Determine for this later time each of the following:
i. The voltmeter reading
ii. The charge on the capacitor
Express all answers in terms of t.
(D) By setting up the appropriate differential equations, prove that the energy stored in the capacitor after a long time is 0.0030 J.
(E) At a still later time t = T, the switch S is moved to position B. Determine the voltmeter reading for some arbitrary time t > T. Express all answers in terms of t.
(F) A long time after t = T, the current in R1 reaches a constant final value If.
i. Determine If.
ii. Use a differential equation to show that the final energy stored in the inductor is 0.45 J.
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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