Conservation Laws in Physics

Why This Matters

Conservation laws are the backbone of physics problem-solving. They let you bypass complicated force analyses and jump straight to answers. On the AP Physics C: Mechanics exam, you're tested on your ability to recognize when a quantity is conserved and why it's conserved in that situation. 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 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 need to account for energy dissipation.

Conservation of Mechanical Energy

• Total mechanical energy $E = K + U$ stays constant when nonconservative forces (friction, air resistance, applied pushes/pulls) do zero work on the system
• The work-energy theorem $\Delta K = W_{net}$ bridges force analysis and energy analysis: the net work done on an object equals its change in kinetic energy
• System boundaries matter. Choosing what's "inside" your system determines whether a force counts as internal (conservative) or external (potentially nonconservative). For example, if you include the Earth in your system, gravity becomes an internal conservative force and you can use $U_g = mgh$. If you don't include the Earth, gravity is an external force doing work on your object.

Energy in Collisions

• Elastic collisions conserve kinetic energy. You use both momentum and energy equations together to solve for unknown velocities.
• Inelastic collisions dissipate kinetic energy into thermal energy, sound, and deformation. Total energy is still conserved when you account for all forms, but mechanical energy is not.
• Perfectly inelastic collisions (objects stick together) lose the maximum kinetic energy possible while still conserving momentum. These are a favorite FRQ scenario because the math is clean: one momentum equation, one unknown final velocity.

Energy in Oscillating Systems

In simple harmonic motion, energy continuously transforms between kinetic and potential forms while the total mechanical energy stays constant (assuming no damping).

• Spring potential energy $U_s = \frac{1}{2}kx^2$ exchanges with kinetic energy $K = \frac{1}{2}mv^2$ as the mass oscillates back and forth
• Gravitational potential energy $U_g = mgh$ provides the restoring mechanism for pendulums; energy oscillates between kinetic and potential just like in a spring system
• Maximum speed occurs at equilibrium (where all energy is kinetic); maximum displacement occurs at the turning points (where all energy is potential and $v = 0$)

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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 like gravity and friction are typically negligible compared to the large internal forces acting over very short time intervals.

Linear Momentum Fundamentals

• Momentum is a vector. You must conserve $p_x$ and $p_y$ components separately in two-dimensional collisions. A common mistake is trying to conserve the magnitude of momentum rather than each component independently.
• The impulse-momentum theorem $\vec{J} = \Delta\vec{p} = \int \vec{F} \, dt$ relates the area under a force-time graph to the change in momentum. On the exam, you may need to estimate this area graphically for non-constant forces.
• Center-of-mass velocity $v_{cm} = \frac{p_{total}}{m_{total}}$ remains constant for an isolated system, even as individual objects change velocities. This is a powerful check on your collision answers.

Collision Types

• Elastic collisions conserve both momentum and kinetic energy. The relative velocity of approach equals the relative velocity of separation (objects bounce apart at the same relative speed they approached with). You need two equations (momentum and energy) to solve for two unknown final velocities.
• Inelastic collisions conserve momentum only. Kinetic energy decreases as some converts to thermal energy or permanent deformation.
• Perfectly inelastic collisions result in objects sticking together. Use $m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_f$ for the simplest momentum calculation, since there's only one unknown.

Explosions and Recoil

Explosions are "reverse collisions." Internal forces separate fragments while total momentum stays constant. If the system starts at rest, total momentum remains zero, meaning the fragments' momenta must sum to zero.

• Recoil problems apply momentum conservation directly: 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 enormous forces on each other during the interaction.

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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, with torque playing the role that force plays in the linear case.
• Point particle angular momentum $L = mvr\sin\theta$ applies when analyzing objects moving in curved paths around a reference point. Here, $r$ is the distance from the reference point and $\theta$ is the angle between $\vec{r}$ and $\vec{v}$. For circular motion, $\theta = 90°$ and this simplifies to $L = mvr$.

Changing Moment of Inertia

• The figure skater effect: pulling mass closer to the rotation axis decreases $I$, so $\omega$ must increase to keep $L$ constant. Quantitatively, if a skater halves their moment of inertia, their angular speed doubles.
• Rotational kinetic energy $K_r = \frac{1}{2}I\omega^2$ is not conserved when $I$ changes. You can see this by substituting $L = I\omega$: since $K_r = \frac{L^2}{2I}$, decreasing $I$ while $L$ stays constant actually increases $K_r$. The skater does internal work (using muscles) to pull their arms in, and that's where the extra kinetic energy comes from.
• Collapsing systems (like a contracting star or gas cloud) spin faster as radius decreases, for the same reason.

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 through the constraint equations $v_{cm} = R\omega$ and $a_{cm} = R\alpha$. These aren't conservation laws themselves, but they're essential for solving problems where both translational and rotational motion are present.

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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: an object slides down a ramp (energy conservation), then collides with another object (momentum conservation), and the combined object flies off a table (projectile motion with energy conservation again).

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 the short interaction time
• Use angular momentum conservation when objects rotate and no external torques act on the system

Multi-Step Problem Strategy

1. Read the full problem first and identify distinct phases (ramp slide, collision, projectile flight, etc.)
2. For each phase, determine which conservation law governs it. Collision phases conserve momentum (not kinetic energy, unless elastic). Motion phases between collisions often conserve mechanical energy.
3. Solve each phase in order, chaining results together. The final velocity from one phase becomes the initial condition for the next.
4. Check your answer. Does the center-of-mass velocity make sense? Did kinetic energy decrease (not increase) in an inelastic collision?

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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?