Conservation Laws in Physics to Know for AP Physics C: Mechanics (2025)

What You Need to Know

Conservation laws let you skip force-by-force dynamics when the right conditions hold. On AP Physics C: Mechanics, the big three are:

• Linear momentum (collisions/explosions, recoil, center of mass motion)
• Mechanical energy (work-energy + conservative forces)
• Angular momentum (rotation, point masses, changing radius, no external torque)

Core idea (the “when can I conserve?” test)

A quantity is conserved when the corresponding external influence is zero for your chosen system:

• Momentum conserved if net external impulse is zero: $\vec J_{\text{ext}} = \int \vec F_{\text{ext}}\,dt = \vec 0$
• Angular momentum conserved about a point/axis if net external torque is zero: $\vec \tau_{\text{ext}} = \frac{d\vec L}{dt} = \vec 0$
• Mechanical energy conserved if only conservative forces do work (or $W_{\text{nc}}=0$): $\Delta(K+U)=0$

Exam mantra: You don’t “choose” a conservation law because it’s convenient; you check the conditions, then use it.

Definitions you must be fluent with

• Momentum: $\vec p = m\vec v$
• System momentum: $\vec P = \sum \vec p_i$
• Impulse: $\vec J = \Delta \vec p$
• Work-energy theorem: $W_{\text{net}} = \Delta K$
• Mechanical energy: $E_{\text{mech}} = K + U$
• Angular momentum (particle about origin): $\vec L = \vec r \times \vec p$
• Angular momentum (rigid body about fixed axis): $L = I\omega$

Step-by-Step Breakdown

A. Momentum conservation problems (collisions/explosions)

1. Choose the system (often both interacting objects). Decide if external forces are negligible during the short interaction time.
2. Check impulse condition: if $\vec J_{\text{ext}} \approx \vec 0$ over the collision/explosion, then $\vec P_i = \vec P_f$.
3. Write momentum conservation in components:
   • $\sum p_{x,i} = \sum p_{x,f}$
   • $\sum p_{y,i} = \sum p_{y,f}$
4. If it’s a collision, decide if you also have energy information:
   • Elastic: also conserve kinetic energy: $K_i = K_f$
   • Inelastic: kinetic energy not conserved; if they stick, use one final velocity.
5. Solve algebraically; keep track of vector directions.

Micro-example (1D perfectly inelastic):

• Two masses stick: $m_1 v_{1i} + m_2 v_{2i} = (m_1+m_2)v_f$

B. Energy conservation / work-energy problems

1. Pick initial and final states (positions + speeds). Decide whether to use mechanical energy or full work-energy.
2. Identify forces that do work and classify:
   • Conservative: gravity, spring.
   • Nonconservative: kinetic friction, applied pushes (usually), drag.
3. Use one of these clean frameworks:
   • Conservative only: $K_i + U_i = K_f + U_f$
   • Include nonconservative work: $K_i + U_i + W_{\text{nc}} = K_f + U_f$
4. Express energies:
   • $K = \tfrac12 mv^2$ (plus rotational if rolling)
   • $U_g = mgy$ (near Earth) or $U_g = -\frac{GMm}{r}$ (universal)
   • $U_s = \tfrac12 kx^2$
5. Solve for the target (often $v$, height, compression $x$).

Micro-example (with friction):

• Sliding distance $d$ with $f_k=\mu_k N$ on level surface: $W_f = -f_k d = -\mu_k mgd$

C. Angular momentum conservation problems

1. Choose the axis/point about which you compute $\vec L$ (this is huge).
2. Check torque condition about that axis/point: if $\vec \tau_{\text{ext}}=\vec 0$ (or negligible), then $\vec L_i=\vec L_f$.
3. Use the appropriate form:
   • Particle: $\vec L = \vec r\times m\vec v$
   • Fixed-axis rotation: $L = I\omega$
4. Be careful: angular momentum is conserved about a specific point/axis, not automatically “in general.”

Micro-example (skater):

• If external torque about vertical axis is negligible: $I_i\omega_i = I_f\omega_f$

Key Formulas, Rules & Facts

Conservation law “trigger” table

Linear momentum and impulse essentials

• Particle momentum: $\vec p = m\vec v$
• System momentum: $\vec P = \sum_i m_i\vec v_i$
• Impulse-momentum theorem: $\vec J = \int \vec F\,dt = \Delta \vec p$
• For constant force: $\vec J = \vec F\Delta t$
• If $\vec F_{\text{ext}}=\vec 0$ then $\frac{d\vec P}{dt}=\vec 0$

Center of mass link (often paired with momentum):

• $M\vec v_{\text{cm}} = \vec P$
• $M\vec a_{\text{cm}} = \sum \vec F_{\text{ext}}$
  So if $\sum \vec F_{\text{ext}}=\vec 0$, then $\vec v_{\text{cm}}$ is constant.

Collision types (what’s conserved?)

Work-energy + potentials (high yield)

• Work-energy theorem: $W_{\text{net}} = \Delta K$
• Conservative force definition: $W_c = -\Delta U$
• Mechanical energy update rule: $\Delta(K+U)=W_{\text{nc}}$
• Gravity near Earth: $U_g = mgy$ and $\Delta U_g = mg\Delta y$
• Universal gravitation: $U_g = -\frac{GMm}{r}$
• Spring potential: $U_s = \tfrac12 kx^2$

Work by common forces:

• Constant force at angle: $W = \vec F\cdot \Delta \vec r = F\Delta r\cos\theta$
• Kinetic friction: $W_f = -f_k d = -\mu_k N d$
• Static friction in pure rolling: often $W_{f_s}=0$ because contact point is instantaneously at rest (but friction can change rotational/translational speeds).

Rotational energy + rolling (conservation-friendly)

• Rotational kinetic energy: $K_{\text{rot}} = \tfrac12 I\omega^2$
• Total kinetic (rolling): $K = \tfrac12 mv^2 + \tfrac12 I\omega^2$
• Rolling without slipping: $v = \omega R$

Angular momentum + torque essentials

• Particle angular momentum: $\vec L = \vec r \times \vec p$ with magnitude $L = r p\sin\theta$
• Rigid body about fixed axis: $L = I\omega$
• Torque: $\vec \tau = \vec r\times \vec F$
• Rotational dynamics: $\sum \tau_{\text{ext}} = \frac{d\vec L}{dt}$

Key “axis choice” fact: if an external force’s line of action passes through your chosen point, its torque about that point is $0$ even if the force is not zero.

Examples & Applications

1) Ballistic pendulum (classic: momentum then energy)

A projectile embeds in a block and the combo swings upward.

Setup:

1. During collision: very short time, external impulse from gravity is negligible, so conserve momentum:
   $m v_{i} = (m+M)V$
2. After collision (swing): mechanical energy conserved (neglect air resistance, pivot friction):
   $\tfrac12 (m+M)V^2 = (m+M)gh$

Key insight: momentum conservation gets you the speed right after impact; energy conservation gets you the rise height.

2) Block + spring with friction (use $W_{\text{nc}}$)

A block with initial speed $v_i$ slides on a rough surface and compresses a spring by $x$.

Setup:

• Choose initial at first contact with spring, final at max compression (speed $0$).
• Use: $K_i + U_i + W_f = K_f + U_f$
• With $U_{s,i}=0$, $K_f=0$:
  $\tfrac12 mv_i^2 - \mu_k mgd = \tfrac12 kx^2$

Key insight: friction is nonconservative; don’t try to hide it inside a potential energy.

3) 2D glancing collision (momentum in components)

Mass $m_1$ moving along +x collides with stationary $m_2$ and they separate at angles.

Setup:

• Use component momentum conservation:
  $m_1 v_{1i} = m_1 v_{1f}\cos\theta_1 + m_2 v_{2f}\cos\theta_2$
  $0 = m_1 v_{1f}\sin\theta_1 - m_2 v_{2f}\sin\theta_2$

Key insight: In 2D you almost always solve using x/y components; do not conserve “speed” or treat momentum as scalar.

4) Person on a turntable pulls in weights (angular momentum)

A person rotates on a frictionless turntable holding masses at radius $r$, then pulls them inward.

Setup:

• External torque about the vertical axis is negligible, so:
  $I_i\omega_i = I_f\omega_f$
• Rotational kinetic energy changes:
  $K_{\text{rot}} = \tfrac12 I\omega^2$

Key insight: $L$ is conserved but $K_{\text{rot}}$ is not necessarily conserved; the person does internal work pulling masses inward.

Common Mistakes & Traps

1. Mixing up “no external force” with “no external impulse.”

   • Wrong: refusing to use momentum in collisions because gravity exists.
   • Fix: check the collision time; if $\Delta t$ is tiny, $\vec J_g = \int m\vec g\,dt \approx \vec 0$.

2. Assuming kinetic energy is conserved in every collision.

   • Wrong: using $K_i=K_f$ for inelastic collisions.
   • Fix: default to momentum conservation; add $K$ conservation only if explicitly elastic (or clearly implied, e.g., ideal billiard balls).

3. Using mechanical energy conservation when friction/drag is present without adding $W_{\text{nc}}$.

   • Wrong: $K_i+U_i=K_f+U_f$ on a rough incline.
   • Fix: write $K_i+U_i+W_{\text{nc}}=K_f+U_f$ with $W_f<0$.

4. Sign errors in potential energy changes.

   • Wrong: writing $\Delta U_g = -mgh$ when the object rises by $h$.
   • Fix: near Earth, $U_g=mgy$; if $y$ increases, $\Delta U_g>0$.

5. Choosing a bad axis for angular momentum conservation.

   • Wrong: conserving $\vec L$ about a point where there is external torque.
   • Fix: choose a point where external forces have zero moment arm (e.g., pivot point) or are negligible.

6. Forgetting rotational kinetic energy in rolling problems.

   • Wrong: using $mgh=\tfrac12 mv^2$ for a rolling disk/sphere.
   • Fix: use $mgh=\tfrac12 mv^2+\tfrac12 I\omega^2$ with $v=\omega R$.

7. Treating momentum conservation as scalar in 2D.

   • Wrong: $m_1 v_{1i} = m_1 v_{1f} + m_2 v_{2f}$ without components.
   • Fix: conserve x and y components separately.

8. Thinking static friction always does zero work (or always does work).

   • Wrong: blanket statements.
   • Fix: in pure rolling on a fixed surface, static friction typically does no work on the rolling object because the contact point doesn’t slide, but it can still change motion via torque; in other setups (moving surfaces), it can do work.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can state exactly when each is conserved:
  • $\vec P$: $\vec J_{\text{ext}}=\vec 0$
  • $\vec L$: $\vec \tau_{\text{ext}}=\vec 0$ (about chosen axis)
  • $K+U$: $W_{\text{nc}}=0$
• In collisions, you default to: $\vec P_i=\vec P_f$ (components in 2D).
• You only use $K_i=K_f$ if the collision is elastic.
• You can write and use: $K_i+U_i+W_{\text{nc}}=K_f+U_f$ without hesitation.
• You remember both gravity potentials: $U_g=mgy$ and $U_g=-\frac{GMm}{r}$.
• For rolling: $K=\tfrac12 mv^2+\tfrac12 I\omega^2$ and $v=\omega R$.
• For angular momentum, you always ask: “About what point/axis is torque zero?”

You’ve got this—pick the right system, check the conservation condition, then let the algebra do the work.