AP Physics 2: Algebra-Based — Cheatsheet
Formulas, exam-day tips, and key terms on one page.
Formulas & relationships
Average translational kinetic energy
KE_avg = (3/2) k_B · T
k_B = 1.38 × 10⁻²³ J·K⁻¹ is Boltzmann’s constant. T must be in kelvin — this relationship only holds on the absolute scale.
Root-mean-square speed
v_rms = √(3 k_B · T / m)
The typical molecular speed grows with √T, not T. Heavier molecules (larger m) move slower at the same temperature.
Ideal gas law
P · V = n · R · T
R = 8.314 J·(mol·K)⁻¹. Use SI units: P in pascals, V in m³, T in kelvin. Then PV has units of joules.
Combined gas law (fixed amount of gas)
P₁V₁ / T₁ = P₂V₂ / T₂
When n is constant, PV/T stays constant. This is the go-to tool for “a gas changes from state 1 to state 2” problems.
Heat and temperature change
Q = m · c · ΔT
c is the specific heat (J·(kg·K)⁻¹): the heat needed to raise 1 kg by 1 K. ΔT = T_final − T_initial, so Q is positive when the object is heated.
First law of thermodynamics
ΔU = Q − W
Q is heat added TO the gas; W is work done BY the gas. Adding heat raises internal energy; letting the gas do work (expand) lowers it.
Thermal efficiency
e = W / Q_H = 1 − (Q_C / Q_H)
Efficiency is the fraction of input heat turned into work. It is always between 0 and 1 (0–100%).
Carnot (maximum) efficiency
e_c = 1 − (T_C / T_H)
The best efficiency any engine can reach between two reservoirs. T_C and T_H must be in kelvin. No real engine beats this ideal limit.
Coulomb’s law
F = k · |q₁ · q₂| / r²
k = 8.99 × 10⁹ N·m²·C⁻² (often rounded to 9.0 × 10⁹). r is the center-to-center distance. The force acts along the line joining the charges: repulsive for like charges, attractive for opposite.
Electric field: definition and point-charge value
E = F / q and E = k · |Q| / r²
Units are N·C⁻¹ (equivalently V·m⁻¹). The first form gives the force felt by a charge q in a field; the second gives the field a source charge Q creates at distance r.
Potential of a point charge & energy of a charge
V = k · Q / r and ΔU = q · ΔV
Keep the sign of Q in V = kQ/r: a positive charge makes V positive, a negative charge makes V negative. ΔU is the energy change when a charge q moves through a potential difference ΔV.
Uniform field and potential
E = V / d
Between two parallel plates a distance d apart with a voltage V across them, the field is uniform with magnitude V/d. This is why field units can be written as V·m⁻¹.
Uniform field between parallel plates
E = V / d
Two large parallel plates with a voltage V across a gap d produce a nearly uniform field V/d, pointing from the positive plate to the negative plate. This is the geometry of a parallel-plate capacitor.
Current, Ohm’s law, and power
I = Q / t · V = I · R · P = I · V = I² · R = V² / R
Power P is the rate at which the resistor converts electrical energy to heat/light, in watts (W). All three power forms are equivalent — pick the one matching your known quantities.
Combining resistors
Series: R = R₁ + R₂ + … · Parallel: 1/R = 1/R₁ + 1/R₂ + …
Series total is bigger than any part; parallel total is smaller than any part. For two parallel resistors, a handy shortcut is R = (R₁R₂)/(R₁ + R₂).
Kirchhoff’s rules
Junction: ΣI_in = ΣI_out · Loop: ΣΔV = 0
Going through a resistor in the direction of current is a voltage drop (−IR); passing from − to + inside a battery is a rise (+EMF). Reverse the sign if you traverse against that direction.
Capacitor charge and RC time constant
Q = C · V · τ = R · C
The time constant τ (tau), in seconds, sets the pace of charging: after one τ the capacitor reaches about 63% of full charge, and after ~5τ it is essentially fully charged. Larger R or C means slower charging.
Field of a long straight wire
B = μ₀ · I / (2π · r)
μ₀ = 4π × 10⁻⁷ T·m·A⁻¹ is the permeability of free space. B is measured in teslas (T). The field is proportional to the current and inversely proportional to the distance r from the wire.
Magnetic force on a charge and on a wire
F = q · v · B · sinθ · F = B · I · L
θ is the angle between v and B. Maximum force at θ = 90° (sinθ = 1); zero force at θ = 0°. For a wire, L is the length within the field and I the current.
Magnetic flux and Faraday’s law
Φ = B · A · cosθ · |EMF| = N · |ΔΦ / Δt|
ΔΦ/Δt is the rate of change of flux. N is the number of turns in the coil. The induced EMF depends on how fast the flux changes, not on the flux itself.
Lenz’s law (the minus sign)
EMF = − N · (ΔΦ / Δt)
The negative sign encodes Lenz’s law: the induced EMF (and current) opposes the change in flux. It is the mathematical statement that induction resists whatever is changing.
Law of reflection
θ_incidence = θ_reflection (both measured from the normal)
A ray hitting the mirror at angle θ to the surface makes an angle (90° − θ) to the normal, and reflects at that same angle on the other side of the normal.
Index of refraction and Snell’s law
n = c / v · n₁ · sinθ₁ = n₂ · sinθ₂
All angles are measured from the normal. The product n·sinθ is the same on both sides of the boundary, so a bigger n forces a smaller angle.
Mirror equation and magnification
1/f = 1/d_o + 1/d_i · m = − d_i / d_o
Sign conventions: d_i is positive for a real image (in front of the mirror), negative for a virtual image (behind). A positive m is upright; negative m is inverted; |m| > 1 is enlarged.
Thin-lens equation and magnification
1/f = 1/d_o + 1/d_i · m = − d_i / d_o
For a converging lens f > 0; for a diverging lens f < 0. A positive d_i is a real image on the far side of the lens; a negative d_i is a virtual image on the same side as the object.
The wave relationship
v = f · λ · f = 1 / T
Wave speed equals frequency times wavelength. In a given medium the speed is fixed by the medium’s properties, so frequency and wavelength trade off inversely.
Wave relationship for sound
v = f · λ (v ≈ 343 m·s⁻¹ in air at room temperature)
The same v = fλ governs sound. A higher-pitched note (larger f) has a shorter wavelength in the same air.
Interference conditions & double slit
Constructive: Δ = m·λ · Destructive: Δ = (m + ½)·λ · d·sinθ = m·λ
Δ is the path difference and m = 0, 1, 2, … is the order. For a double slit of spacing d, bright fringes appear at angles where d·sinθ = mλ.
Diffraction grating (bright lines)
d · sinθ = m · λ
Same form as the double slit, where d is the spacing between adjacent slits and m is the order. Smaller slit spacing d spreads the orders farther apart.
Photon energy & the photoelectric equation
E = h · f · KE_max = h · f − φ
φ is the work function — the minimum energy to free an electron from the metal. Electrons are emitted only when hf ≥ φ; the excess energy becomes the electron’s maximum kinetic energy.
Photon energy from a transition
E_photon = E_high − E_low = h · f
The emitted or absorbed photon carries exactly the energy difference between the two levels. A larger energy gap means a higher-frequency (shorter-wavelength) photon.
Half-life & mass–energy
remaining fraction = (1/2)ⁿ, n = t / t₁/₂ · E = m · c²
The half-life t₁/₂ is the time for half of a sample to decay; after n half-lives, a fraction (1/2)ⁿ remains. E = mc² converts the mass lost in a reaction into released energy.
de Broglie wavelength & photon momentum
λ = h / p · p_photon = h / λ
h = 6.63 × 10⁻³⁴ J·s. Wavelength and momentum are inversely related: heavy, fast particles have vanishingly small wavelengths; light, slow particles have larger ones.
On the exam
- Two proportionalities to lock in: KE_avg ∝ T (linear in absolute temperature), but v_rms ∝ √T. When a problem doubles the temperature, energy doubles while speed rises only by √2.
- When a problem gives “before” and “after” states, reach for P₁V₁/T₁ = P₂V₂/T₂ and cancel whatever is held constant. Constant V → P ∝ T; constant T → P ∝ 1/V; constant P → V ∝ T.
- Read the wording carefully: “work done ON the gas” is the opposite sign of the W in ΔU = Q − W (which is work done BY the gas). Compression → gas does negative work → U tends to rise; expansion → gas does positive work → U tends to fall.
- Compare the two efficiencies: e = 1 − Q_C/Q_H uses the actual heat flows, while e_c = 1 − T_C/T_H is the temperature-set ceiling. A real engine’s efficiency is always less than its Carnot limit — if a problem’s engine beats Carnot, you have an error.
- When only ratios change (charge or distance scaled up or down), you rarely need to plug into Coulomb’s law fully. Track the proportionality: F ∝ q₁q₂/r². Double one charge → ×2; triple the distance → ×1/9.
- Two different formulas share the letter E. Use E = F/q when a charge and the force on it are given; use E = kQ/r² when a source charge and a distance are given. Mixing them up is a classic slip.
- Remember the different distance dependences: the field of a point charge falls off as 1/r², but its potential falls off as 1/r. Energy questions use potential (scalar, add algebraically); force questions use the field (vector).
- Lock in the conductor facts: field zero inside, excess charge on the surface, field perpendicular just outside, and charge densest at sharp points. These appear on nearly every electrostatics free-response question.
- Know all three power formulas: P = IV, P = I²R, and P = V²/R. If a problem gives you resistance and voltage but not current, P = V²/R saves a step. They are algebraically identical via Ohm’s law.
- Anchor each rule with its shared quantity: series shares current, parallel shares voltage. From that, use V = IR branch by branch. Complex networks reduce to one equivalent resistor by collapsing series and parallel groups one step at a time.
- Pair the rules with their conservation law: junction rule = conservation of charge, loop rule = conservation of energy. On multi-loop problems, write one junction equation and one loop equation per unknown current, then solve the system.
- For RC problems, answer three questions: what happens at t = 0 (capacitor = wire), what happens as t → ∞ (capacitor = open, fully charged with Q = CV), and how fast (τ = RC). Those three checkpoints capture nearly every exam item.
- Two facts anchor this topic: magnetism comes from moving charge, and magnetic field lines are always closed loops (no monopoles). If a diagram shows field lines starting or stopping in mid-space, something is wrong.
- For any magnetic-force problem, first find the angle between v and B. Parallel → no force; perpendicular → maximum force qvB. The direction always comes from a right-hand rule, remembering to flip it for a negative charge.
- Faraday’s law depends on the *rate* of flux change, not the flux itself. A huge steady field induces nothing; a small field that changes quickly can induce a large EMF. Always compute ΔΦ/Δt, and multiply by N for a coil.
- To get an induced-current direction: (1) decide whether flux is increasing or decreasing, (2) the induced field opposes that change, (3) use the right-hand rule to find the current direction that makes that field. Then sanity-check with energy conservation — the loop should resist the motion.
- Memorize the plane-mirror image: virtual, upright, same size, equal distance behind. It is the reference point you compare curved-mirror and lens images against later in the unit.
- Total internal reflection has two requirements, and both must hold: (1) light moving from high n to low n, and (2) an angle of incidence beyond the critical angle. If either fails, some light refracts through and it is not total.
- Convex mirrors and diverging lenses share a signature: they always make virtual, upright, diminished images, no matter where the object is. Concave mirrors and converging lenses are the versatile ones whose image type depends on object position.
- Learn the converging-lens cases by object position (beyond 2f, at 2f, between f and 2f, inside f) — the exam tests all four. And remember the shortcut: diverging lenses always give virtual, upright, diminished images, no computation needed.
- When a wave passes from one medium to another, its *frequency stays the same* (set by the source), while its speed and wavelength both change. This fact underlies refraction: light slows in glass, so its wavelength shortens while its color (frequency) is unchanged.
- Doppler shorthand: approaching → higher pitch (compressed waves), receding → lower pitch (stretched waves). The dramatic drop you hear as a vehicle passes is the switch from approaching to receding at the instant it goes by.
- Convert everything to metres before using d sinθ = mλ: millimetres are 10⁻³ and nanometres are 10⁻⁹. A power-of-ten error here is the most common way to lose the double-slit point.
- Sort the evidence: interference and diffraction demonstrate light’s *wave* nature; the photoelectric effect and Compton scattering demonstrate its *particle* nature. Light is both — the exam expects you to name which experiment reveals which side.
- Separate the two knobs: frequency controls each electron’s energy (KE_max = hf − φ), while intensity controls the *number* of electrons. Exam questions constantly test whether you know that brighter light does not make faster electrons.
- Line spectra are the fingerprint of quantization. Each element’s unique set of energy gaps gives it a unique pattern of spectral lines — the reason spectroscopy can identify the composition of distant stars.
- For half-life problems, first find n = t / t₁/₂, then the surviving fraction is (½)ⁿ. Watch that decay is exponential, not linear — after 2 half-lives one-quarter remains, not zero.
- Two duality relations to keep straight: for a photon E = hf and p = h/λ; for a matter particle λ = h/p. Both hinge on Planck’s constant — the bridge between the wave quantities (f, λ) and the particle quantities (E, p).
How to get a 5
- In circuit problems, redraw the circuit if it looks confusing. Identify nodes to clearly see what's in series and what's in parallel.
- For right-hand rule questions involving electrons (negative charges), use your right hand and flip the result, or use your left hand.
- Remember that work done BY a gas is positive when it expands, meaning work done ON the gas is negative (W = -PΔV).
- In optics, draw ray diagrams carefully. Real images are formed by converging rays and can be projected; virtual images cannot.
Key terms
Ideal Gas Law — PV = nRT = NkT. Relates pressure, volume, temperature, and amount of gas.
First Law of Thermodynamics — ΔU = Q + W (where W is work done ON the gas). Energy conservation for thermal systems.
Electric Field of a Point Charge — E = k|q|/r². Points radially outward from positive, inward toward negative.
Electric Potential (Voltage) — V = U_E / q. The electric potential energy per unit charge. V = kq/r for a point charge.
Capacitance Formula — C = Q/V. For a parallel-plate capacitor: C = ε₀A/d.
Resistors in Series and Parallel — Series: R_eq = R₁ + R₂ + ... Parallel: 1/R_eq = 1/R₁ + 1/R₂ + ...
Magnetic Force on a Moving Charge — F_B = qvB sinθ. Direction given by the Right-Hand Rule.
Faraday's Law of Induction — ε = -N (ΔΦ_B / Δt). Induced EMF is proportional to the rate of change of magnetic flux.
Index of Refraction — n = c/v. Ratio of the speed of light in vacuum to the speed of light in the material.
Snell's Law — n₁ sinθ₁ = n₂ sinθ₂. Describes refraction (bending of light) at an interface.
Energy of a Photon — E = hf = hc/λ. Energy is quantized and depends only on frequency (or wavelength).
Mass-Energy Equivalence — E = mc². Mass can be converted to energy and vice versa.