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

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.