Conservation Laws in Physics

Why This Matters

Conservation laws are the backbone of physics problem-solving—they're the shortcuts that let you bypass complicated force analyses and jump straight to answers. On the AP Physics C: Mechanics exam, you're being tested on your ability to recognize when a quantity is conserved and why it's conserved in that situation. The key insight? Conservation laws emerge from the absence of external influences: no net external force means momentum is conserved, no net external torque means angular momentum is conserved, and no work by nonconservative forces means mechanical energy is conserved.

These principles connect directly to the exam's focus on system selection, energy transfer, and collision analysis. Whether you're analyzing a perfectly inelastic collision, a figure skater's spin, or a mass-spring oscillator, you'll need to identify which conservation law applies and justify your reasoning. Don't just memorize that "momentum is conserved in collisions"—know that it's conserved because internal forces cancel by Newton's third law and external impulses are negligible during the short interaction time.

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Energy Conservation: The Universal Bookkeeper

Energy conservation is perhaps the most versatile tool in mechanics. The total mechanical energy $E = K + U$ remains constant when only conservative forces do work. The moment nonconservative forces enter the picture, you'll need to account for energy dissipation.

Conservation of Mechanical Energy

• Total mechanical energy $E = K + U$ stays constant—but only when nonconservative forces (friction, air resistance) do zero work on the system
• Work-energy theorem $\Delta K = W_{net}$ connects force analysis to energy analysis; the net work done equals the change in kinetic energy
• System boundaries matter—choosing what's "inside" your system determines whether a force is internal (conservative) or external (potentially nonconservative)

Energy in Collisions

• Elastic collisions conserve kinetic energy—use both momentum and energy equations together to solve for unknown velocities
• Inelastic collisions dissipate kinetic energy to thermal energy and sound, but total energy is still conserved when you account for all forms
• Perfectly inelastic collisions (objects stick together) lose the maximum kinetic energy while still conserving momentum—a favorite FRQ scenario

Energy in Oscillating Systems

• Spring potential energy $U_s = \frac{1}{2}kx^2$ exchanges continuously with kinetic energy in simple harmonic motion
• Gravitational potential energy $U_g = mgh$ provides the restoring mechanism for pendulums; energy oscillates between kinetic and potential
• Maximum speed occurs at equilibrium where all energy is kinetic; maximum displacement occurs where all energy is potential

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Momentum Conservation: Collision Analysis Powerhouse

Linear momentum $\vec{p} = m\vec{v}$ is conserved whenever the net external impulse on a system is zero. During collisions, external forces are typically negligible compared to the large internal forces acting over short time intervals.

Linear Momentum Fundamentals

• Momentum is a vector—you must conserve $p_x$ and $p_y$ components separately in two-dimensional collisions
• Impulse-momentum theorem $\vec{J} = \Delta\vec{p} = \int \vec{F} \, dt$ relates the area under a force-time graph to momentum change
• Center-of-mass velocity $v_{cm} = \frac{P_{total}}{M_{total}}$ remains constant for isolated systems—even as individual objects change velocities

Collision Types

• Elastic collisions conserve both momentum and kinetic energy; relative velocity reverses direction (objects bounce apart at the same relative speed)
• Inelastic collisions conserve momentum only; kinetic energy decreases as some converts to thermal energy or deformation
• Perfectly inelastic collisions result in objects sticking together; use $m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$ for the simplest momentum calculation

Explosions and Recoil

• Explosions are "reverse collisions"—internal forces separate fragments while total momentum remains zero (or constant if initially moving)
• Recoil problems apply momentum conservation: a gun firing a bullet, a person jumping off a cart, or rocket propulsion all follow $\vec{p}_{initial} = \vec{p}_{final}$
• Internal forces cancel by Newton's third law—this is why momentum is conserved even when objects exert huge forces on each other

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Angular Momentum Conservation: Rotational Dynamics

Angular momentum $\vec{L} = I\vec{\omega}$ is conserved when no net external torque acts on a system. This principle governs everything from spinning skaters to orbiting planets.

Angular Momentum Fundamentals

• Angular momentum $L = I\omega$ depends on both moment of inertia and angular velocity; change one and the other adjusts to compensate
• No external torque means $L_i = L_f$—this is the rotational analog of linear momentum conservation
• Point particle angular momentum $L = mvr\sin\theta$ applies when analyzing objects moving in curved paths around a reference point

Changing Moment of Inertia

• Figure skater effect—pulling mass closer to the rotation axis decreases $I$, so $\omega$ must increase to keep $L$ constant
• Rotational kinetic energy $K_r = \frac{1}{2}I\omega^2$ is not conserved when $I$ changes; the skater does internal work to speed up
• Collapsing systems (like a contracting star) spin faster as radius decreases—angular momentum conservation explains why

Rotational Collisions

• Objects coupling together (like a person jumping onto a merry-go-round) conserve angular momentum: $I_1\omega_1 + I_2\omega_2 = I_{total}\omega_f$
• Rotational inelastic collisions lose rotational kinetic energy just like linear inelastic collisions lose translational kinetic energy
• Rolling without slipping connects linear and angular motion: $v_{cm} = R\omega$ and $a_{cm} = R\alpha$

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Connecting Conservation Laws: When Multiple Apply

Many AP problems require you to recognize which conservation law applies at each stage of a multi-step scenario. The classic example: a collision followed by projectile motion, or an object sliding down a ramp before colliding with another object.

Choosing the Right Law

• Use energy conservation when you need to relate speeds at different positions and no nonconservative forces act
• Use momentum conservation during collisions or explosions where external forces are negligible over short time intervals
• Use angular momentum conservation when objects rotate and no external torques act on the system

Multi-Step Problem Strategy

• Break the problem into phases—identify which conservation law governs each phase separately
• Collision phases conserve momentum (not energy, unless elastic); motion phases between collisions often conserve energy
• Final answers often require chaining results—the final velocity from one phase becomes the initial condition for the next

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Quick Reference Table

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Self-Check Questions

1. A block slides down a frictionless ramp and collides with a stationary block on a frictionless surface. Which conservation law applies during the slide down the ramp, and which applies during the collision? Why does the answer differ?

2. Two objects undergo an elastic collision. Which two conservation equations would you use to solve for both final velocities, and why is one equation insufficient?

3. Compare a figure skater pulling in their arms to two ice skaters pushing apart from rest. Which quantities are conserved in each scenario, and which are not?

4. An FRQ shows a force-time graph for a collision. How would you determine the impulse delivered, and what does this tell you about the change in momentum?

5. A rotating platform has a person standing at its edge who then walks toward the center. Explain why angular momentum is conserved but rotational kinetic energy is not. Where does the "extra" energy come from?