Velocity with constant acceleration
v = v0 + at
v final velocity, v0 initial velocity, a acceleration, t time
Use when time is given and distance isn't needed.
Practice 1 question →Equations to know by heart
The MCAT doesn't give you a formula sheet. These 73 equations come up again and again. Each links to questions that make you apply it.
Start a timed equations blockKinematics and forces
Displacement with constant acceleration
Δx = v0t + ½at2
Δx displacement, v0 initial velocity, a acceleration, t time
From rest, this reduces to Δx = ½at². Free fall uses a = g.
Practice 3 questions →Velocity–displacement relation
v2 = v02 + 2aΔx
v final velocity, v0 initial velocity, a acceleration, Δx displacement
The go-to equation when no time is given, such as stopping distances.
Practice 1 question →Newton's second law
Fnet = ma
Fnet net force, m mass, a acceleration
Sum all forces first. On an incline, gravity's component along the ramp is mg sin θ.
Practice 3 questions →Friction
f = μN
f friction force, μ coefficient of friction (static or kinetic), N normal force
On a flat surface with no vertical forces besides gravity, N = mg.
Practice 2 questions →Centripetal acceleration and force
ac = v2/r · Fc = mv2/r
v speed, r radius, m mass
Centripetal force isn't a new force; friction, tension, gravity, or a normal force supplies it.
Practice 4 questions →Newton's law of gravitation
F = Gm1m2/r2
G gravitational constant, m1, m2 masses, r center-to-center distance
Inverse square: triple the distance, one-ninth the force. Measure r from the planet's center.
Practice 2 questions →Torque
τ = rF sin θ
r lever arm, F force, θ angle between r and F
In rotational equilibrium, clockwise torques equal counterclockwise torques.
Practice 2 questions →Work, energy, momentum, and oscillation
Work
W = Fd cos θ
F force, d displacement, θ angle between them
Forces perpendicular to motion (normal force, magnetic force) do no work. Friction does negative work.
Practice 1 question →Power
P = W/t = ΔE/t
W work, ΔE energy change, t time
Lifting at constant speed: P = mgh/t.
Practice 1 question →Kinetic and gravitational potential energy
KE = ½mv2 · PE = mgh
m mass, v speed, g gravitational acceleration, h height
With no friction, mgh = ½mv², so v = √(2gh) regardless of mass. Doubling speed quadruples KE.
Practice 4 questions →Springs (Hooke's law)
F = −kx · U = ½kx2
k spring constant, x displacement from equilibrium
Force is linear in x; stored energy goes as x².
Practice 1 question →Momentum and impulse
p = mv · J = FΔt = Δp
p momentum, J impulse, F average force, Δt contact time
Momentum is conserved in every collision; kinetic energy only in elastic ones. Longer Δt means smaller force.
Practice 4 questions →Pendulum period
T = 2π√(L/g)
T period, L length, g gravitational acceleration
Mass doesn't matter. Quadrupling the length doubles the period.
Practice 3 questions →Fluids and gases
Density and specific gravity
ρ = m/V · SG = ρ/ρwater
ρ density, m mass, V volume; ρwater = 1 g/cm³ = 1,000 kg/m³
SG above 1 sinks in water; for a floating object, SG = fraction submerged.
Practice 3 questions →Pressure and Pascal's principle
P = F/A · F1/A1 = F2/A2
P pressure, F force, A area
Convert area carefully: 1 cm² = 10⁻⁴ m².
Practice 2 questions →Hydrostatic pressure
P = P0 + ρgh
P0 surface pressure, ρ fluid density, h depth
Gauge pressure is ρgh alone. About 10 m of water adds 1 atm.
Practice 2 questions →Buoyant force (Archimedes)
Fb = ρfluidVsubg
ρfluid fluid density, Vsub submerged volume
Buoyant force = weight in air − apparent weight in fluid.
Practice 2 questions →Continuity equation
A1v1 = A2v2
A cross-sectional area (πr²), v flow speed
Halving the radius cuts area to ¼, so speed rises 4×.
Practice 2 questions →Bernoulli's equation
P + ½ρv2 + ρgh = constant
P pressure, ρ density, v speed, h height
Horizontal flow: faster means lower pressure. Use v₂² − v₁², not (v₂ − v₁)².
Practice 2 questions →Poiseuille's law
Q = πr4ΔP/(8ηL)
Q flow rate, r radius, ΔP pressure difference, η viscosity, L length
Flow scales with r⁴: halving the radius cuts flow to 1/16.
Practice 2 questions →Ideal gas law
PV = nRT · P1V1/T1 = P2V2/T2
P pressure, V volume, n moles, R gas constant (0.0821 L·atm/mol·K), T temperature in kelvin
Always use kelvin. At STP, 1 mol occupies 22.4 L.
Practice 3 questions →Dalton's law of partial pressures
PA = XAPtotal
PA partial pressure, XA mole fraction
Partial pressures add up to the total.
Practice 1 question →Graham's law of effusion
r1/r2 = √(M2/M1)
r effusion rate, M molar mass
Lighter gases effuse faster, by the square root of the mass ratio.
Practice 1 question →Electricity and magnetism
Coulomb's law and electric field
F = kq1q2/r2 · E = kq/r2
k = 9 × 10⁹ N·m²/C², q charge, r distance
Inverse square, like gravity. Fields from several charges add as vectors.
Practice 2 questions →Electric potential and potential energy
V = kq/r · W = qΔV
V potential, q charge, r distance, W work
Potential falls off as 1/r (not 1/r²) and is a scalar. 1 V = 1 J/C.
Practice 2 questions →Uniform field between plates
E = V/d
E field strength, V potential difference, d plate separation
Convert d to meters.
Practice 1 question →Ohm's law
V = IR
V voltage, I current, R resistance
Find total resistance first, then total current, then each branch.
Practice 2 questions →Electrical power
P = IV = I2R = V2/R
P power, I current, V voltage, R resistance
In parallel (same V), the smaller resistor uses more power.
Practice 2 questions →Resistors in series and parallel
Rs = R1 + R2 · 1/Rp = 1/R1 + 1/R2
Rs series total, Rp parallel total
A parallel total is always smaller than the smallest branch.
Practice 3 questions →Capacitance and stored energy
C = Q/V = κε0A/d · U = ½CV2
C capacitance, Q charge, κ dielectric constant, A plate area, d separation
Capacitors combine opposite to resistors: add in parallel, reciprocals in series.
Practice 4 questions →Magnetic force on a moving charge
F = qvB sin θ · r = mv/(qB)
q charge, v speed, B field, θ angle between v and B, r radius of circular path
Zero force when moving parallel to the field. The force does no work.
Practice 3 questions →Waves, sound, and light
Wave speed
v = fλ · T = 1/f
v wave speed, f frequency, λ wavelength, T period
Frequency is set by the source; the medium sets speed, so λ changes between media.
Practice 1 question →Sound level
β = 10 log(I/I0)
β sound level in dB, I intensity, I0 threshold intensity
Each 10× intensity adds 10 dB. Doubling distance from a point source quarters intensity (about −6 dB).
Practice 2 questions →Doppler effect
f′ = f (v ± vo)/(v ∓ vs)
f′ observed frequency, v speed of sound, vo observer speed, vs source speed
Use the signs that raise f′ when source and observer approach.
Practice 1 question →Standing waves
λ = 2L/n (both ends same) · λ = 4L/n, n odd (one end closed)
λ wavelength, L length, n harmonic number
Strings and open pipes: all harmonics. Closed pipes: odd harmonics only.
Practice 2 questions →Beat frequency
fbeat = |f1 − f2|
f1, f2 the two frequencies
The perceived pitch is the average; the beat rate is the difference.
Practice 1 question →Photon energy
E = hf = hc/λ
h Planck's constant, f frequency, c speed of light, λ wavelength; hc ≈ 1,240 eV·nm
Shorter wavelength, more energy: violet beats red.
Practice 3 questions →Index of refraction and Snell's law
n = c/v · n1 sin θ1 = n2 sin θ2
n index of refraction, v speed in the medium, θ angle from the normal
Into higher n, light slows and bends toward the normal.
Practice 1 question →Critical angle
sin θc = n2/n1
θc critical angle, n1 starting medium (higher n), n2 second medium
Total internal reflection only from higher to lower n.
Practice 1 question →Thin lens and mirror equation
1/f = 1/o + 1/i · m = −i/o · f = R/2
f focal length, o object distance, i image distance, m magnification, R radius of curvature
Positive i: real and inverted. Negative i: virtual and upright.
Practice 4 questions →Lens power
P = 1/f
P power in diopters, f focal length in meters
Converging lenses are positive, diverging negative. Lenses in contact add.
Practice 1 question →Atoms and nuclei
Hydrogen energy levels
En = −13.6 eV/n2
En energy of level n
Photon energy = difference between two levels.
Practice 1 question →Photoelectric effect
KEmax = hf − Φ
KEmax maximum electron energy, hf photon energy, Φ work function
Below threshold (hf less than Φ), no electrons at any intensity.
Practice 3 questions →Radioactive decay
N = N0(½)t/t½
N amount remaining, N0 starting amount, t elapsed time, t½ half-life
Count halvings: 1/8 remaining means 3 half-lives.
Practice 2 questions →Mass–energy equivalence
E = mc2
m mass in kg, c = 3 × 10⁸ m/s
Convert grams to kilograms before squaring c.
Practice 1 question →Solutions, acids, and equilibrium
pH and pOH
pH = −log[H+] · pH + pOH = 14
[H+] hydrogen ion concentration (M), at 25 °C
For a × 10⁻ⁿ, pH = n − log a. A larger coefficient lowers the pH a little.
Practice 3 questions →Henderson–Hasselbalch
pH = pKa + log([A−]/[HA])
pKa of the acid, [A−] conjugate base, [HA] acid
Equal amounts: pH = pKa. Each 10× ratio shifts pH by 1.
Practice 1 question →pH of a weak acid
[H+] ≈ √(KaC)
Ka acid dissociation constant, C initial concentration
Valid when the acid dissociates only slightly.
Practice 1 question →Solubility product
Ksp = [Mn+]a[Xm−]b
ion concentrations at saturation, raised to their coefficients
MX: Ksp = s². MX₂: Ksp = 4s³. A common ion lowers solubility.
Practice 1 question →Molarity and dilution
M = mol/L · M1V1 = M2V2
M molarity, V volume
V₂ is the final volume, not the water added. Serial dilutions multiply.
Practice 2 questions →Colligative properties
ΔT = iKm · Π = iMRT
i van 't Hoff factor, K freezing or boiling constant, m molality, Π osmotic pressure, M molarity
NaCl gives i = 2, CaCl₂ gives i = 3, glucose gives i = 1.
Practice 3 questions →Equilibrium constant and reaction quotient
K = [C]c[D]d/[A]a[B]b
concentrations at equilibrium (K) or at any moment (Q); leave out pure solids and liquids
Q less than K: forward. Q greater than K: reverse. Only temperature changes K.
Practice 2 questions →Rate law
rate = k[A]m[B]n
k rate constant, m and n reaction orders (from experiments or the slow step)
Triple a second-order reactant: rate ×9. Halve a first-order one: rate ×½.
Practice 2 questions →Thermodynamics and electrochemistry
Heat transfer
q = mcΔT · q = mL
m mass, c specific heat, ΔT temperature change, L latent heat
During a phase change, temperature stays constant: use q = mL.
Practice 3 questions →First law of thermodynamics
ΔU = Q − W
ΔU internal energy change, Q heat added, W work done by the system
Isothermal ideal gas: ΔU = 0, so Q = W.
Practice 2 questions →Gibbs free energy
ΔG = ΔH − TΔS
ΔH enthalpy change, T temperature (K), ΔS entropy change
Match units: ΔS is usually in J/K, ΔH in kJ.
Practice 3 questions →Free energy and the equilibrium constant
ΔG° = −RT ln K
R = 8.314 J/mol·K, T temperature (K), K equilibrium constant
K = 1 means ΔG° = 0. At 298 K, each 10× in K is about 5.7 kJ/mol.
Practice 2 questions →Enthalpy from bond energies
ΔH = Σ bonds broken − Σ bonds formed
bond enthalpies in kJ/mol
Opposite of the products-minus-reactants rule used for heats of formation.
Practice 1 question →Standard cell potential
E°cell = E°cathode − E°anode
standard reduction potentials of each half-cell
Don't multiply potentials by stoichiometric coefficients.
Practice 1 question →Free energy and cell potential
ΔG° = −nFE°
n moles of electrons, F = 96,500 C/mol, E° cell potential
Positive E° means negative ΔG°: spontaneous (galvanic).
Practice 1 question →Electrolysis (Faraday's law)
mol e− = It/F
I current (A), t time (s), F = 96,500 C/mol
Divide moles of electrons by the ion's charge to get moles of metal.
Practice 1 question →Nernst equation
E = (61.5 mV/z) log([ion]out/[ion]in)
z ion charge, concentrations outside and inside the cell, at 37 °C
K⁺ is about −90 mV, Na⁺ about +60 mV. A divalent ion halves the factor.
Practice 2 questions →Biochemistry and physiology
Michaelis–Menten
v = Vmax[S]/(Km + [S])
v reaction velocity, [S] substrate concentration, Km [S] at half Vmax
At [S] = Km, v = ½Vmax. Competitive inhibitors raise apparent Km.
Practice 3 questions →Turnover number and catalytic efficiency
kcat = Vmax/[E]T · efficiency = kcat/Km
[E]T total enzyme concentration
Watch time units when converting to s⁻¹.
Practice 2 questions →Hardy–Weinberg
p + q = 1 · p2 + 2pq + q2 = 1
p, q allele frequencies; 2pq heterozygote frequency
Start from q² = the recessive phenotype frequency.
Practice 3 questions →Isoelectric point
pI = (pKa1 + pKa2)/2
the two pKa values that bracket the neutral form
Acidic amino acids: average the two lowest. Basic: the two highest.
Practice 3 questions →Cardiac output and blood pressure
CO = HR × SV · MAP ≈ CO × TPR
HR heart rate, SV stroke volume, MAP mean arterial pressure, TPR total peripheral resistance
Changes multiply: double TPR and cut CO by 25%, MAP rises 1.5×.
Practice 2 questions →Renal clearance
C = (U × V)/P
U urine concentration, V urine flow rate, P plasma concentration
Clearance above GFR means net secretion; below means net reabsorption.
Practice 2 questions →Psych/Soc, statistics, and epidemiology
Weber's law
ΔI/I = k
ΔI just-noticeable difference, I starting intensity, k Weber fraction
The just-noticeable difference is a constant fraction, not a constant amount.
Practice 1 question →z-score
z = (x − μ)/σ
x score, μ mean, σ standard deviation
In a normal distribution, z = 1 is about the 84th percentile; ±1 SD covers about 68%.
Practice 2 questions →Relative risk
RR = riskexposed / riskunexposed
risk = cases / people in the group
Risk difference (attributable risk) subtracts instead of divides.
Practice 1 question →Sensitivity, specificity, and predictive values
Sens = TP/(TP + FN) · Spec = TN/(TN + FP) · PPV = TP/(TP + FP)
TP true positives, FN false negatives, TN true negatives, FP false positives
Sensitivity and specificity don't change with prevalence; PPV and NPV do.
Practice 3 questions →