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Conservation of Energy

AP Course

AP Physics C: Mechanics

Updated for the AP Physics C: Mechanics framework effective fall 2024, with calculus-based derivations, labs, quizzes, and practice.

Build calculus-based mechanics with original derivations, experiments, quizzes, and practice sets.

AP Physics C: Mechanics · Unit 3 · Topic 3.4

Conservation of Energy

Line integrals, potential functions, and power track energy transfer when forces vary with position or time.

Exponential change · Conservation of EnergyA constant fractional change produces a curved exponential trend rather than a constant absolute decrease.

1. Topic Lens

Conservation of Energy is studied through work, energy, and power. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[W=\int_{x_i}^{x_f}F_x(x)\,dx\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. Use F=ma and the chain rule a=v dv/dx.
  2. Then F dx=mv dv, so accumulated work becomes an integral over speed.
  3. Evaluating the bounds gives the change in kinetic energy.
\[W_{\text{net}}=\int mv\,dv=\frac12m(v_f^2-v_i^2)\]

3. Detailed Visual Model

Conservation of EnergyTopic-matched fine pixel-art model · use the lesson's equations and labeled diagrams for exact scientific meaning.

4. Worked Example and Lab Link

A one-dimensional force is F(x)=2x N from x=0 to 3 m. Find its work.

Answer: W=∫₀³2x dx=[x²]₀³=9 J.

Investigation idea: Use a force sensor to obtain a nonconstant F-x curve and compare numerical area with measured ΔK.

Common trap: For a conservative force, F=-dU/dx; missing the minus sign reverses stability conclusions.

Checkpoint · Topic 3.4

Explain how conservation of energy supports or limits this conclusion: W=∫₀³2x dx=[x²]₀³=9 J.

Official curriculum reference: College Board AP Physics C: Mechanics course page. The explanation and worked example are independently written for this study site.