AP Physics 2 Quick Review

Temperature is a measure of the average translational kinetic energy of a system's particles; gas pressure is the aggregate result of countless particle collisions with a container's walls.

Average kinetic energy per particle: $KE_avg=/32k_BT$ (T in kelvin only), with $k_B=1.38×10^-23$ J/K. RMS speed: $v_rms=sqrt/3k_BTm=sqrt/3RTM$.

Temperature (average KE per particle, intensive, independent of amount) vs. internal/thermal energy (total KE of all particles, extensive) --- doubling the amount of gas at the same T doubles internal energy but does NOT change temperature.

Plugging a Celsius temperature directly into $KE_avg=/32k_BT$ or any gas law --- always convert to kelvin ($T_K=T_C+273$) before using any kinetic-theory or gas-law formula.

$KE_avg=/32k_BT$, always kelvin; higher T means faster average molecular speed, not more particles.

The ideal gas law treats gas particles as point masses with no intermolecular forces, linking pressure, volume, amount, and temperature in one equation.

$PV=nRT=Nk_BT$, with $R=8.31$ J/(mol$·$K). At constant $n$: $/P_1V_1T_1=/P_2V_2T_2$. Isothermal: $PV=$ constant; isobaric: $V/T=$ constant; isochoric: $P/T=$ constant.

$n$ (moles, paired with $R$) vs. $N$ (number of particles, paired with $k_B$) --- mixing the two constants misses a factor of Avogadro's number, $N_A=6.02×10^23$/mol.

Forgetting to convert volume to m$^3$, pressure to Pa, and temperature to kelvin before using $PV=nRT$ --- always convert to consistent SI units and absolute temperature first.

$PV=nRT$ (kelvin, SI units); combined gas law handles changing states at fixed $n$.

Heat flows spontaneously from higher to lower temperature until thermal equilibrium is reached, via conduction, convection, or radiation.

Thermal equilibrium: no net heat flow, equal temperatures. Conduction rate: $/Qt=/kA TL$. Calorimetry (no losses): heat lost by the hot object equals heat gained by the cold object.

Heat $Q$ (energy transferred BETWEEN objects due to a temperature difference) vs. temperature $T$ (a state variable) vs. internal energy $U$ (total energy contained) --- heat is a transfer process, not a property an object "has."

In calorimetry, dropping the sign or mixing up which $ T$ is final-minus-initial for the hot vs. cold object --- define $ T=T_f-T_i$ consistently for each substance so heat lost is negative and heat gained is positive.

Heat flows hot$→$cold until $T_final$ is equal; conduction rate $ kA T/L$.

Energy conservation applied to thermal systems: the change in a gas's internal energy equals the heat added minus the work the gas does on its surroundings.

$ U=Q-W$ ($W=$ work done BY the gas, $W=P V$ at constant pressure). Isochoric: $W=0$, $ U=Q$. Isothermal (ideal gas): $ U=0$, $Q=W$. Adiabatic: $Q=0$, $ U=-W$. On a $PV$ diagram, work done by the gas equals the area under the curve.

Work done BY the gas (positive when the gas expands, subtracts from $ U$) vs. work done ON the gas (opposite sign) --- AP Physics 2 uses $ U=Q-W_by\ gas$.

Sign errors from mixing the two competing work conventions --- stay consistent, and remember that gas expansion ($ V>0$) means the gas does POSITIVE work, which removes energy from the gas.

$ U=Q-W_by$; expansion $arrow$ gas does $+$ work $arrow$ removes energy from the gas.

Different materials require different amounts of heat to raise their temperature (specific heat), and phase changes absorb or release heat with no change in temperature.

Sensible heat: $Q=mc T$. Latent heat (phase change): $Q=mL$, with $ T=0$ during the change. A heating curve plateaus (flat) during melting/boiling and slants during single-phase heating.

Specific heat $c$ (governs the sloped, temperature-changing regions, J/(kg$·$K)) vs. latent heat $L$ (governs the flat phase-change plateau, J/kg) --- these are two different formulas for two different segments of a heating curve.

Applying a single $Q=mc T$ across a range that includes a phase change (e.g., ice to steam) --- break the process into separate heating and phase-change segments and sum the $Q$ for each.

Heating curve: slanted segment $=mc T$; flat plateau $=mL$ (phase change, $T$ constant).

Entropy measures a system's disorder/energy dispersal; the total entropy of an isolated system never decreases, and heat spontaneously flows only from hot to cold.

$ S=/QT$ (T in kelvin) for a reversible process at constant temperature. Second Law: total entropy of an isolated system (system $+$ surroundings) always increases or, ideally, stays constant --- never decreases.

A specific system's entropy CAN decrease (e.g., water freezing) --- the Second Law requires only that the TOTAL entropy of system $+$ surroundings increases for a spontaneous process.

Assuming entropy must increase for every process examined in isolation --- check whether the question asks about the system alone or the system plus surroundings; a local decrease is fine if compensated elsewhere.

$ S=Q/T$ (kelvin); total entropy of an isolated system only increases or stays the same.

Like charges repel and opposite charges attract via Coulomb's law, an inverse-square force analogous in form to gravity but far stronger and able to be either attractive or repulsive.

$F=k/|q_1q_2|r^2$, with $k=8.99×10^9$ N$·$m$^2$/C$^2$ $≤ft(k=/14_0)$. Charge is quantized ($q=ne$, $e=1.6×10^-19$ C) and always conserved in a closed system.

Coulomb's law force (a vector, sign-dependent, can attract or repel) vs. Newtonian gravitation (always attractive) --- despite the identical $1/r^2$ form, don't drop the direction reasoning unique to electric force.

Forgetting Coulomb's law force is a VECTOR --- with three or more charges, find each pairwise force vector and add them by components (superposition), not just add magnitudes.

$F=kq_1q_2/r^2$; superpose force VECTORS for multiple charges, don't just add magnitudes.

Net electric charge in an isolated system is always conserved; objects can be charged by friction, conduction (contact), or induction (no contact).

Conduction: direct contact, charge transfers, both objects end with the SAME sign. Induction: a charged object induces charge separation in a nearby neutral conductor without contact; grounding it and removing the ground while the charge is nearby leaves the OPPOSITE sign.

Charging by conduction (contact, same-sign result) vs. charging by induction (no contact, requires a temporary ground, opposite-sign result) --- a very common mix-up.

Assuming induction transfers charge FROM the inducing object like conduction does --- in induction, the final charge on the neutral object is OPPOSITE in sign to the inducing charge, and no direct charge transfer from that object occurs.

Conduction $=$ contact, same sign; induction $=$ no contact $+$ ground, opposite sign.

An electric field is the force per unit charge that a small positive test charge would feel at each point in space, existing whether or not a test charge is actually there.

$E=/Fq=k/|Q|r^2$ (point charge); direction points away from $+$, toward $-$. Field lines never cross, start on $+$ charges, end on $-$ charges. Uniform field (parallel plates): $E=/Vd$.

Electric field $E$ (force per unit charge, a vector property of space) vs. electric force $F=qE$ (the actual force on a specific charge placed in that field).

Using the sign of the source charge instead of a POSITIVE test charge to determine field direction --- the field around a negative charge always points TOWARD it, even though the source charge itself is negative.

$E=kQ/r^2$; direction $=$ away from $+$, toward $-$; $F=qE$ gives the actual force on a placed charge.

Electric potential energy is the work required to assemble a configuration of charges (or move a charge within a field) against the electric force, analogous to gravitational PE.

$U=k/q_1q_2r$ (two point charges, $U=0$ at $r=∞$). Work by the field: $W_by field=- U=U_i-U_f$. For a charge crossing a potential difference $ V=V_f-V_i$: $W_by field=-q V$.

Like charges: $U>0$ (repel, PE drops as they separate) vs. opposite charges: $U<0$ (attract, PE becomes more negative --- more bound --- as they approach).

Dropping the sign of the charges when computing $U=kq_1q_2/r$ --- $U$ is negative for one $+$ and one $-$ charge, meaning positive work must be ADDED to separate them to infinity.

$U=kq_1q_2/r$; like charges: $U>0$; opposite charges: $U<0$ (bound state).

Electric potential (voltage) is potential energy per unit charge, a scalar property of a point in space that avoids vector force analysis when finding energy changes.

$V=/Uq=k/Qr$ ($V=0$ at $r=∞$). In a uniform field: $V=Ed$. Equipotential surfaces are always perpendicular to field lines; no work is done moving a charge along an equipotential.

Electric potential $V$ (scalar, add contributions from multiple charges ALGEBRAICALLY with sign) vs. electric field $E$ (vector, must be added as COMPONENTS) --- potential superposition is simple scalar addition.

Treating potential from multiple point charges as a vector sum like the field, instead of simple signed scalar addition --- this mistake makes an easy problem needlessly hard (and wrong).

$V$ is scalar (add algebraically); $E$ is vector (add as components); $V=Ed$ in a uniform field.

A capacitor stores charge (and energy) by separating opposite charges on two conductors, with capacitance a fixed geometric property relating stored charge to voltage.

$C=/QV$; parallel plate: $C=/_0Ad$. Energy stored: $U=/12QV=/12CV^2=/Q^22C$. Series: $/1C_eq=/1C_i$. Parallel: $C_eq= C_i$.

Capacitors combine OPPOSITELY from resistors --- series capacitors add as reciprocals (equivalent $C$ is smaller than the smallest), parallel capacitors add directly (equivalent $C$ is larger).

Applying resistor combination rules directly to capacitors --- always remember capacitor rules are inverted relative to resistors.

Capacitors: series $arrow$ reciprocals add (opposite of resistors); parallel $arrow$ direct sum. $U=/12CV^2$.

Total mechanical plus electric potential energy is conserved for a charge moving through a field with no other forces doing work, enabling energy methods instead of kinematics.

$KE_i+U_i=KE_f+U_f$, or $/12mv_i^2+qV_i=/12mv_f^2+qV_f$. Work by the field on a charge crossing potential difference $ V$: $W=-q V$.

A positive charge released from rest accelerates toward LOWER potential (loses PE, gains KE) vs. a negative charge, which accelerates toward HIGHER potential.

Assuming all charges accelerate toward lower potential --- true only for positive charges; negative charges (like electrons) speed up moving toward higher potential, a frequent sign error with charged particles between plates.

$+$ charge "falls" toward lower $V$; $-$ charge "falls" toward higher $V$; $ KE=-q V$.

Conventional current is the rate of flow of positive charge, defined to flow from high to low potential in the external circuit even though electrons (the actual carriers in metals) move the opposite way.

$I=/ Q t$, unit ampere (A) $=$ C/s. Conventional current direction $=$ direction positive charge would flow. Drift velocity relation: $I=nqv_dA$.

Conventional current direction (defined as positive-charge flow, used in all circuit diagrams/equations) vs. actual electron flow (opposite direction) --- always use conventional current for analysis.

Confusing the slow drift velocity (mm/s) of individual electrons with the near-instant propagation of the electrical signal through the circuit --- a bulb lights almost instantly even though electrons barely move.

Conventional current $=$ direction of $+$ charge flow (opposite actual electron motion); $I= Q/ t$.

How resistors are wired determines whether current or voltage is the shared quantity, and dictates how their resistances combine.

Series: same current through each element, $R_eq= R_i$, voltages add. Parallel: same voltage across each branch, $/1R_eq=/1R_i$, currents add.

Series resistors (add directly, $R_eq$ always LARGER than any individual $R$) vs. parallel resistors (reciprocals add, $R_eq$ always SMALLER than the smallest individual $R$).

Forgetting to invert at the end when combining parallel resistors --- computing $/1R_eq=/1R_1+/1R_2$ and reporting that sum as $R_eq$ instead of flipping it.

Series: $R$'s add, same $I$. Parallel: $1/R$'s add (then invert!), same $V$.

Ohm's law relates voltage, current, and resistance for ohmic materials; resistance itself is a geometric/material property governed by resistivity, length, and cross-sectional area.

$V=IR$. $R=/LA$ ($=$ resistivity). Resistivity of most conductors increases with temperature (more collisions with vibrating lattice ions).

Resistance $R$ (depends on material AND geometry/size) vs. resistivity $$ (an intrinsic material property, independent of a wire's length or thickness).

Doubling a wire's diameter and assuming $R$ is halved --- $R1/A$, and $A d^2$, so doubling the diameter QUARTERS the resistance, not halves it.

$R= L/A$; longer wire $=$ more $R$; thicker wire (bigger $A$) $=$ less $R$, scales with $d^2$, not $d$.

Electric power is the rate at which electrical energy converts to another form (heat, light, mechanical work) in a circuit element.

$P=IV=I^2R=/V^2R$. Energy dissipated: $E=Pt$. Battery power delivered: $P=IE$; power lost internally: $I^2r$.

$P=I^2R$ vs. $P=V^2/R$ --- both are correct but only simultaneously valid via $V=IR$ for the SAME resistor; don't mix values from different circuit elements.

Using $P=V^2/R$ with the battery's full EMF instead of the voltage actually across a specific resistor in a multi-resistor circuit --- always match the $V$, $I$, and $R$ of the SAME element.

$P=IV=I^2R=V^2/R$ --- use matching $I$, $V$, $R$ for the SAME element.

Most real circuits combine series and parallel sections; solving them requires systematically reducing the network to a single equivalent resistance, then working backward.

Method: (1) reduce parallel/series groups step by step to $R_eq$, (2) find total current $I=E/(R_eq+r)$ (include internal resistance $r$ if given), (3) work backward to find each branch's current/voltage.

Ammeters (measure current, near-zero resistance, placed IN SERIES) vs. voltmeters (measure voltage, near-infinite resistance, placed IN PARALLEL) --- inserting either the wrong way alters the circuit's actual behavior.

Forgetting a battery's internal resistance $r$ when given --- terminal voltage is $V_terminal=E-Ir$, less than the EMF whenever current flows.

Reduce to $R_eq$ $arrow$ find total $I$ $arrow$ work backward; terminal $V=E-Ir$.

Kirchhoff's two rules are statements of energy conservation (loop rule) and charge conservation (junction rule), needed for circuits that can't be reduced by simple series/parallel combination.

Loop Rule: $ V=0$ around any closed loop (gains at EMF sources, drops across resistors in the assumed current direction). Junction Rule: $ I_in= I_out$ at any node.

Loop rule (energy conservation, around a closed path) vs. junction rule (charge conservation, at a node) --- most multi-loop circuits require BOTH, applied enough times to solve for all unknowns.

Getting confused by a negative current after solving --- a negative result simply means the actual current flows opposite your assumed direction; the magnitude is still correct, so don't "fix" the sign.

Loop: $ V=0$ (energy). Junction: $I_in=I_out$ (charge). Negative $I=$ wrong assumed direction, not an error.

In a resistor-capacitor circuit, current and voltage change smoothly and exponentially over time as the capacitor charges or discharges, governed by the time constant $RC$.

Charging: $V_C(t)=E(1-e^-t/RC)$. Discharging: $V_C(t)=V_0e^-t/RC$. Time constant $=RC$: after one $$, charging reaches $≈63%$ of final value; discharging falls to $≈37%$ of initial value.

Right after a switch closes (uncharged capacitor acts like a SHORT, max current flows) vs. long after (steady state, fully charged capacitor acts like an OPEN switch, current through that branch $=0$).

Assuming current keeps flowing through a fully charged capacitor branch --- in steady state, NO current flows through a capacitor branch; it behaves as an open circuit.

$t=0$: capacitor $=$ wire (max $I$). $t→∞$: capacitor $=$ open switch ($I=0$ in that branch). $=RC$.

Magnetic fields are produced by moving charges (currents) and permanent magnets, represented by field lines that always form closed loops rather than beginning or ending on isolated poles.

Field lines point from N to S pole outside a magnet (S to N inside). Unlike electric field lines, magnetic field lines have NO beginning or end --- no magnetic monopoles exist. Unit: tesla (T).

Magnetic field lines (always closed loops, no isolated N or S pole) vs. electric field lines (begin on $+$ charges, end on $-$ charges/infinity, isolated charges do exist).

Assuming a bar magnet cut in half yields an isolated single pole --- cutting a magnet always produces two new complete magnets, each with its own N and S pole.

B-field lines form closed loops (N to S outside); no magnetic monopoles exist, unlike isolated electric charges.

A magnetic field exerts a force on a moving charge only when velocity has a component perpendicular to the field, with direction found via the right-hand rule.

$F=qvB$; direction via right-hand rule (fingers along $v$, curl toward $B$, thumb gives $F$ for a $+$ charge). This force is always $$ to $v$, doing NO work, producing uniform circular motion with $r=/mvqB$.

Magnetic force (always $$ to velocity, changes direction only, never changes speed/KE) vs. electric force (can point any direction relative to velocity, can speed up or slow down a charge).

Forgetting to reverse the right-hand-rule result for a NEGATIVE charge --- the rule gives the force direction for a $+$ charge; for a $-$ charge, the actual force points exactly opposite.

$F=qvB$, $$ to $v$ always $arrow$ no work, circular motion, $r=mv/qB$. Flip direction for $-$ charges.

Since current is moving charge, a current-carrying wire in an external magnetic field experiences a force --- the basis of motors and of wires pushing or pulling on each other.

$F=BIL$; direction via right-hand rule (fingers along $I$, curl toward $B$, thumb gives $F$). Parallel wires: currents in the SAME direction attract; OPPOSITE directions repel.

Force on a moving point charge ($F=qvB$) vs. force on a current-carrying wire ($F=BIL$) --- same right-hand-rule logic, but the wire formula uses current $I$ and length $L$ instead of charge $q$ and speed $v$.

Mixing up the attract/repel rule for parallel wires with bar-magnet intuition --- derive it (each wire's circular field, via right-hand rule, acts on the other wire) rather than memorizing "like currents attract" blindly.

$F=BIL$; parallel wires: same direction $arrow$ attract, opposite direction $arrow$ repel.

A changing magnetic flux through a loop induces an EMF (and current, if closed), with Lenz's law fixing the direction to oppose the change causing it.

Flux: $=BA$. Faraday's Law: $E=-N/ t$. Lenz's Law: induced current opposes the CHANGE in flux, consistent with energy conservation.

Flux $$ itself (can be large but constant, inducing zero EMF) vs. the RATE OF CHANGE of flux (what actually induces an EMF) --- a stationary loop in a large but unchanging field has no induced current.

Assuming induced current opposes the flux itself rather than the CHANGE in flux --- if flux is increasing, induced current opposes that increase; if flux is decreasing, induced current tries to maintain it (same direction as the original flux).

$E=-N/ t$; induced current opposes the CHANGE in flux, not the flux itself.

Light reflecting off a smooth surface obeys the law of reflection, with angles measured from the normal to the surface, never from the surface itself.

Law of reflection: $_i=_r$, both measured from the NORMAL. Specular reflection (smooth surface, sharp image) vs. diffuse reflection (rough surface, scattered rays, object still visible from many angles).

Angles measured from the NORMAL (perpendicular to the surface) --- a very common habit is measuring from the surface itself, which is always WRONG.

Measuring the angle of incidence/reflection from the reflecting surface instead of the normal --- always draw the dashed normal line first before labeling any angle.

$_i=_r$, ALWAYS from the normal, never from the surface itself.

Curved mirrors form images via the mirror equation, with the signs of image distance and magnification revealing the image's location, orientation, and real/virtual nature.

$/1d_o+/1d_i=/1f$, $f=/R2$. Magnification: $m=-/d_id_o=/h_ih_o$. Concave: $f>0$; convex: $f<0$. $d_i>0=$ real image (same side, in front); $d_i<0=$ virtual (behind mirror). $m>0=$ upright; $m<0=$ inverted.

Concave mirror ($f>0$, converging, can form real OR virtual images) vs. convex mirror ($f<0$, diverging, ALWAYS forms a virtual, upright, reduced image regardless of object position).

Assigning the wrong sign to $f$ for convex vs. concave, which cascades into wrong signs for $d_i$ and $m$ --- memorize "concave $=$ converging $=$ positive $f$; convex $=$ diverging $=$ negative $f$" first.

$1/d_o+1/d_i=1/f$; concave: $f>0$ (converges); convex: $f<0$ (always virtual/upright/smaller).

Light bends when passing between media of different optical densities because its speed changes, with the bending direction related to the index of refraction by Snell's law.

$n=/cv$ ($n≥q1$). Snell's Law: $n_1_1=n_2_2$. Into a denser medium (higher $n$): bends TOWARD the normal, slows down. Total internal reflection: $_c=/n_2n_1$, only possible going from higher to lower $n$.

Total internal reflection (only possible going from HIGH $n$ to LOW $n$, e.g., glass to air, beyond the critical angle) --- it can NEVER occur going from a lower-$n$ to a higher-$n$ medium.

Applying the critical-angle formula when light travels from a less dense to a more dense medium --- always confirm $n_1>n_2$ before even considering total internal reflection.

$n_1_1=n_2_2$; into higher $n→$ bends toward normal, slows; TIR only high $n→$ low $n$, beyond $_c$.

Converging and diverging lenses form images via the same thin-lens equation as mirrors, but light actually passes THROUGH the lens, flipping which side counts as "real."

$/1d_o+/1d_i=/1f$. Converging (convex) lens: $f>0$; diverging (concave) lens: $f<0$ (always). $d_i>0=$ real image (OPPOSITE side from object); $d_i<0=$ virtual (SAME side). $m=-d_i/d_o$.

For MIRRORS, real images form on the SAME side as the object; for LENSES, real images form on the OPPOSITE side --- a frequent sign-convention mix-up when switching between devices.

Reusing a mirror's "same side $=$ real" rule on a lens problem, or vice versa --- always re-check which optical device is in play before interpreting the sign of $d_i$.

Lens: $1/d_o+1/d_i=1/f$; real image $=$ OPPOSITE side from object (mirrors: real $=$ SAME side).

Waves transfer energy without transferring matter, classified as transverse (perpendicular oscillation) or longitudinal (parallel/compression oscillation).

$v=f$. Transverse: particle motion $$ to travel direction (light, string waves). Longitudinal: particle motion parallel to travel direction (sound). String wave speed: $v=sqrtT/$ --- depends on the MEDIUM, not frequency or amplitude.

Wave speed (fixed by the medium's properties) vs. frequency (set by the SOURCE) --- changing the source's frequency does NOT change wave speed; it changes wavelength instead.

Assuming a higher-frequency source makes a wave travel faster through a given medium --- speed is fixed by the medium; frequency changes only affect wavelength ($=v/f$).

$v=f$; $v$ set by the MEDIUM only; changing $f$ (the source) changes $$, not $v$.

Sound is a longitudinal mechanical wave requiring a medium, with pitch and loudness governed independently by frequency and intensity.

Sound speed depends on the medium (fastest in solids, slowest in gases; $≈340$ m/s in air at room temperature). Intensity: $I=/PA/1r^2$ from a point source. Decibel level: $=10≤ft(/II_0)$.

Pitch (perceived, corresponds to FREQUENCY) vs. loudness (perceived, corresponds to INTENSITY/amplitude) --- these are independent wave properties.

Assuming sound travels through a vacuum, or that intensity falls off linearly with distance instead of as $1/r^2$ from a point source.

Sound needs a medium (no vacuum travel); $I1/r^2$; pitch $=$ frequency, loudness $=$ intensity.

What happens when a wave reaches a boundary depends on whether the boundary is fixed or free, and only transverse waves can be polarized by filtering out all but one oscillation direction.

Fixed end: reflected pulse INVERTS (180$^$ phase shift). Free end: reflected pulse does NOT invert. Polarization applies only to TRANSVERSE waves; a polarizer passes only the oscillation component aligned with its transmission axis; two crossed polarizers block all light.

Reflection at a fixed boundary (pulse inverts) vs. a free boundary (pulse does not invert) --- going from less dense to more dense medium behaves like a more "fixed" boundary.

Forgetting that only TRANSVERSE waves (like light) can be polarized --- sound, being longitudinal, cannot be polarized at all.

Fixed end $arrow$ pulse inverts; free end $arrow$ no inversion. Only TRANSVERSE waves can be polarized.

Electromagnetic waves are self-propagating oscillations of electric and magnetic fields requiring NO medium, all traveling at the speed of light in vacuum, differing only in frequency/wavelength.

$c=f$, $c=3.00×10^8$ m/s in vacuum. Spectrum, increasing frequency: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray. $E$ and $B$ fields are $$ to each other AND to the direction of propagation.

Mechanical waves (sound, water --- REQUIRE a medium) vs. electromagnetic waves (light, radio, X-rays --- do NOT require a medium, travel through vacuum).

Assuming different EM waves travel at different speeds in vacuum --- ALL electromagnetic waves travel at exactly $c$ in vacuum regardless of frequency; only $f$ and $$ differ.

All EM waves: same speed $c$ in vacuum; only $f$ and $$ differ; $E B$ direction of travel.

The observed frequency of a wave shifts when source and/or observer move relative to each other --- higher when approaching, lower when receding.

$f_obs=f_source≤ft(/v_wave± v_observerv_wave v_source)$. Choose signs so $f_obs$ increases when approaching and decreases when receding.

The Doppler shift (a change in OBSERVED frequency due to relative motion) vs. an actual change in the source's emitted frequency or the wave's propagation speed --- neither of those ever actually changes.

Swapping the plus/minus signs for approaching vs. receding motion --- reason physically first ("getting closer $arrow$ frequency up; farther apart $arrow$ frequency down") and choose signs to match.

Approaching $arrow$ higher observed $f$; receding $arrow$ lower observed $f$; wave speed itself never changes.

When waves overlap, their displacements add algebraically (superposition); under the right conditions this produces stable standing waves with fixed nodes and antinodes.

Constructive: path difference $=m$. Destructive: path difference $=(m+/12)$. String fixed at both ends: $_n=/2Ln$, $f_n=/nv2L$, all integers $n$. Tube closed at one end: $_n=/4Ln$, ODD $n$ only.

Both-ends-fixed/open boundary (all harmonics, $_n=2L/n$) vs. one-end-closed boundary (odd harmonics only, $_n=4L/n$) --- always identify the boundary conditions first.

Applying the both-ends-fixed harmonic formula to a closed-pipe problem, which only supports ODD harmonics with a different formula --- check the boundary conditions before choosing a formula.

Both ends fixed/open: all harmonics, $_n=2L/n$. One end closed: odd harmonics only, $_n=4L/n$.

Waves bend around obstacles and spread through openings (diffraction), most noticeably when the opening is comparable in size to the wavelength; a grating uses many slits to produce sharp interference maxima.

Diffraction is significant when slit width $≈$. Double-slit/grating bright-fringe condition: $d=m$ ($d=$ slit spacing). More slits $arrow$ sharper, narrower, brighter maxima at the same angular positions.

Diffraction (bending/spreading through a single opening or around an edge) vs. interference (superposition of two or more distinct coherent sources) --- a double-slit pattern actually involves both simultaneously.

Using the single-slit condition when the problem gives slit SPACING (double-slit/grating) instead of slit WIDTH (single-slit diffraction) --- read carefully which quantity is given.

$d=m$ for bright fringes (grating/double-slit); more slits $=$ sharper maxima, same positions.

Light reflecting off the top and bottom of a thin film interferes constructively or destructively depending on the path-length difference AND any phase shifts from reflection.

A 180$^$ phase shift occurs only on reflection off a HIGHER-index medium than the one light is traveling in; no shift reflecting off a lower-index medium. Path difference inside the film $=2t$ (near-normal incidence); use $_film=_0/n_film$.

A phase shift occurs going from LOW-to-HIGH index at a reflection but NOT going from HIGH-to-LOW index --- check BOTH the top and bottom surface of the film separately.

Forgetting to check for a reflection phase shift at both surfaces, or forgetting to convert to the wavelength IN THE FILM before applying the path-difference condition.

Phase shift only low $n→$ high $n$ reflection; check BOTH surfaces; path diff $=2t$; use $_film=_0/n$.

Light and matter both exhibit wave-like and particle-like behavior depending on the experiment used to observe them --- neither description alone is complete.

Photon energy: $E=hf=/hc$, $h=6.63×10^-34$ J$·$s. de Broglie wavelength of matter: $=/hp=/hmv$. Particle evidence: photoelectric effect, Compton scattering. Wave evidence: diffraction, interference.

Wave-like behavior (diffraction, interference) vs. particle-like behavior (discrete energy packets, momentum transfer) --- both are always "true" simultaneously; which is OBSERVED depends on the experiment.

Assuming only light shows wave behavior or only matter shows particle behavior --- $=h/p$ applies to ALL matter, and light behaves as discrete photons in the photoelectric/Compton effects.

$E=hf=hc/$ (photon); $=h/p$ (matter wave) --- duality applies to BOTH light and matter.

In the Bohr model, electrons orbit only in specific quantized energy levels, and atoms emit or absorb photons of exactly the energy corresponding to transitions between those levels.

Hydrogen energy levels: $E_n=-/13.6 eVn^2$ ($n=1$ is ground state). Transition photon: $E_photon=|E_n_f-E_n_i|=hf$. Emission: bright discrete lines (electron drops). Absorption: dark lines at the SAME wavelengths (electron jumps up).

Emission spectrum (bright lines, electrons falling to LOWER levels) vs. absorption spectrum (dark lines, electrons jumping to HIGHER levels) --- both occur at the exact same set of wavelengths for a given element.

Forgetting the negative-energy convention for bound states and computing a transition with the wrong sign, or forgetting the ground state ($n=1$) is the MOST negative energy, not zero.

$E_n=-13.6eV/n^2$; emission $=$ drop $=$ bright line; absorption $=$ jump $=$ dark line, same wavelengths.

All objects emit a continuous electromagnetic spectrum due to their temperature, with peak wavelength and total power both changing predictably with temperature --- a phenomenon that motivated quantum theory.

Wien's Law: hotter objects peak at SHORTER wavelength, $_peak1/T$. Stefan-Boltzmann Law: total radiated power $ T^4$. Planck's hypothesis (energy is quantized) resolved the classical "ultraviolet catastrophe."

A blackbody spectrum is CONTINUOUS (all wavelengths, peak set by temperature alone) vs. an atomic emission spectrum, which is DISCRETE (sharp lines set by the element's energy levels).

Assuming radiated power scales linearly with temperature --- it scales as $T^4$, so a modest temperature increase causes a dramatically larger increase in total emitted power.

Hotter $arrow$ shorter peak $$ (Wien) and power $ T^4$ (Stefan-Boltzmann); blackbody spectrum is continuous, not discrete lines.

Light above a threshold frequency ejects electrons from a metal instantly, with the ejected electrons' maximum kinetic energy depending only on frequency, not intensity --- direct evidence for photons.

$KE_max=hf-$ ($=$ work function). Threshold: $f_0=/h$; below $f_0$, no electrons are ejected at ANY intensity. Increasing intensity (above threshold) increases the NUMBER of ejected electrons, not their $KE_max$; increasing frequency increases $KE_max$.

Light intensity (affects the NUMBER of photoelectrons/current) vs. light frequency (affects the MAXIMUM KINETIC ENERGY of each electron) --- two completely independent effects.

Assuming that enough intensity below the threshold frequency will eventually eject electrons --- no amount of intensity below $f_0$ causes ejection; only photon energy (frequency) determines whether ejection happens.

$KE_max=hf-$; intensity $arrow$ MORE electrons (current); frequency $arrow$ FASTER electrons ($KE_max$). Below $f_0$: nothing, ever.

When a high-energy photon scatters off a free electron, energy and momentum are conserved as if the photon were a particle, so the scattered photon always loses energy and gains wavelength.

$='-=/hm_ec(1-)$, with $/hm_ec=2.43×10^-12$ m. The scattered photon ALWAYS has a longer wavelength (lower energy) than the incident photon.

Compton scattering (photon collides with and scatters off a free electron, wavelength INCREASES, demonstrates photon momentum $p=h/$) vs. the photoelectric effect (photon is fully absorbed, electron ejected, no scattered photon remains).

Assuming the scattered photon's wavelength could stay the same or decrease --- since the electron gains kinetic energy in the collision, the photon MUST lose energy, so $≥q0$ always.

$=/hm_ec(1-)$; scattered photon always has longer $$ (lost energy to the electron).

Unstable nuclei release energy by splitting (fission), combining (fusion), or spontaneously emitting particles/radiation (decay), always converting a small amount of mass into a large amount of energy via $E=mc^2$.

$E=mc^2$; energy released $= m c^2$ (mass defect). Fission: heavy nucleus splits into lighter, more tightly bound nuclei. Fusion: light nuclei combine into a heavier, more tightly bound nucleus. Alpha decay: emits $^4_2He$ (mass number $-4$, atomic number $-2$). Beta-minus decay: neutron $→$ proton $+$ electron $+$ antineutrino (atomic number $+1$). Gamma decay: emits a photon (no change in $A$ or $Z$). Half-life: $N=N_0≤ft(/12)^t/T_1/2$.

Fission (splitting HEAVY nuclei) vs. fusion (combining LIGHT nuclei) --- both release energy because they move nuclei TOWARD iron/nickel, the most tightly bound nuclei on the curve of binding energy per nucleon.

Forgetting to conserve mass number ($A$) and atomic number ($Z$) SEPARATELY when balancing a nuclear equation --- the totals of $A$ must match on both sides, and the totals of $Z$ must match on both sides, independently.

Fission $=$ split heavy; fusion $=$ combine light; both release $E$ by moving toward max binding energy. Always balance $A$ and $Z$ separately.

One formula sheet, six units: gas law, Coulomb/circuit trio, $F=qvB$/$BIL$, mirror-lens equation, $E=hf$.

When two terms sound alike, ask: scalar or vector? same side or opposite side? number or energy?

Read the scenario for its physical setup first, THEN pick the matching formula --- don't formula-hunt blind.

These appear on the official exam equation/table sheet --- know their VALUES cold so you never hunt for them under time pressure.

Diagram $arrow$ identify what's conserved $arrow$ equation per step $arrow$ solve symbolically $arrow$ justify in words.

42 MCQ (all single-select, 85 min, 50%) + 4 FRQ --- math routine, representation translation, experimental design, qual/quant translation (95 min, 50%).

Hypothesis $arrow$ identify variables $arrow$ replicable procedure $arrow$ linearize $arrow$ address uncertainty $arrow$ interpret the slope.

Linearize first, read the slope's units to identify the constant, best-fit line always, error bars overlap $=$ maybe not significant.