Chapter 13: High-Yield Cheat Sheet

This is the MCAT physics cheat sheet you wish you had the week before test day. Every equation, constant, and sign convention from all 12 chapters, compressed into one page. Kinematics, fluids, circuits, optics, thermodynamics, the Doppler effect, thin-lens sign rules, Bernoulli’s equation, the full set of decay rules - every formula, none of the prose. The MCAT does not give you a formula sheet. This page is what you memorize so test day feels like filling in a table you already know.

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1. Math and Data Skills

You will not get a calculator on the MCAT. The physics you do must be fast, mostly by approximation, and almost always with scientific notation. Know these cold before you even open a physics passage.

SI base units: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd). Everything else is derived.

Prefix	Symbol	Multiplier
tera	T	10¹²
giga	G	10⁹
mega	M	10⁶
kilo	k	10³
centi	c	10⁻²
milli	m	10⁻³
micro	μ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²

Scientific notation rules (for test-day mental math):

• Multiply: multiply the coefficients, add the exponents. $(3 \times 10^4)(2 \times 10^3) = 6 \times 10^7$
• Divide: divide the coefficients, subtract the exponents. $(6 \times 10^8) / (3 \times 10^2) = 2 \times 10^6$
• Power: raise coefficient to the power, multiply the exponent. $(2 \times 10^3)^2 = 4 \times 10^6$

Logarithms (base 10 unless stated):

• $\log(ab) = \log a + \log b$
• $\log(a/b) = \log a - \log b$
• $\log(a^n) = n \log a$
• $\log 1 = 0$, $\log 10 = 1$, $\log 100 = 2$
• $\log 2 \approx 0.30$, $\log 3 \approx 0.48$, $\log 5 \approx 0.70$, $\log 7 \approx 0.85$
• pH shortcut: if $[\mathrm{H}^+] = n \times 10^{-x}$, then $\mathrm{pH} = x - \log n$

Natural log: $\ln x = 2.303 \log x$; $\ln e = 1$, $\ln 1 = 0$. Useful for half-life and RC circuits.

Trig values you must memorize:

Angle	sin	cos	tan
0°	0	1	0
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$ ≈ 0.87	$\frac{1}{\sqrt{3}}$ ≈ 0.58
45°	$\frac{\sqrt{2}}{2}$ ≈ 0.71	$\frac{\sqrt{2}}{2}$ ≈ 0.71	1
60°	$\frac{\sqrt{3}}{2}$ ≈ 0.87	$\frac{1}{2}$	√3 ≈ 1.73
90°	1	0	undefined

Small-angle approximations (for θ in radians, θ < ~15°): $\sin\theta \approx \theta$, $\cos\theta \approx 1$, $\tan\theta \approx \theta$.

Vector basics: A vector has magnitude and direction. Resolve any vector $\vec{V}$ at angle θ from horizontal into components:
$V_x = V\cos\theta, \qquad V_y = V\sin\theta$
$|\vec{V}| = \sqrt{V_x^2 + V_y^2}, \qquad \theta = \tan^{-1}(V_y/V_x)$

Dot product: $\vec{A}\cdot\vec{B} = AB\cos\theta$ → scalar. Cross product: $|\vec{A}\times\vec{B}| = AB\sin\theta$ → vector perpendicular to both (right-hand rule).

Dimensional analysis: cancel units across every step. If the final units do not match the answer-choice units, you made an algebra error. MCAT distractors are often off by a factor of 10 or a unit conversion (cm vs m, g vs kg).

Error and uncertainty: for $Q = A \cdot B$ or $Q = A/B$, relative errors add: $\Delta Q/Q = \Delta A/A + \Delta B/B$. For $Q = A + B$, absolute errors add: $\Delta Q = \Delta A + \Delta B$. Significant figures: final answer carries the fewest sig figs of any input.

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2. Kinematics

One-dimensional motion with constant acceleration - the Big Five equations. $x_0, v_0$ = initial, $x, v$ = final, $t$ = time, $a$ = acceleration.

Free fall (no air resistance): $a = g = 9.8~\mathrm{m/s^2}$ downward (use 10 m/s² for quick math). Object dropped from rest: $v = gt$, $h = \tfrac{1}{2} g t^2$.

Projectile motion: horizontal and vertical components are independent. Horizontal: constant velocity, $a_x = 0$, $x = v_{0x} t$. Vertical: $a_y = -g$, $y = v_{0y} t - \tfrac{1}{2} g t^2$.

Projectile Quantity	Formula
Time of flight (level ground)	$t = 2v_0 \sin\theta / g$
Range (level ground)	$R = v_0^2 \sin(2\theta) / g$
Max height	$h = v_0^2 \sin^2\theta / (2g)$
Max range angle	45° (equal launch/land height)

At the peak of a projectile’s path, vertical velocity = 0 but horizontal velocity is unchanged. Symmetric trajectory: speed at launch = speed at landing (same height).

Average velocity for constant acceleration: $\bar{v} = (v_0 + v)/2$.

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3. Forces and Dynamics (Newton’s Laws)

Law	Statement	Key Idea
1st (Inertia)	Object at rest stays at rest; object in motion stays in motion, unless acted on by a net force	Requires an inertial frame
2nd	$\vec{F}_{\text{net}} = m\vec{a}$	Units: 1 N = 1 kg·m/s²
3rd	For every action there is an equal and opposite reaction	Forces act on different objects

Weight: $W = mg$ (always toward Earth). Normal force $N$ is perpendicular to the contact surface, not always equal to weight.

Friction:

• Static: $f_s \le \mu_s N$ (adjusts up to a maximum to prevent motion)
• Kinetic: $f_k = \mu_k N$ (constant once sliding, $\mu_k < \mu_s$)

Inclined plane (angle θ with horizontal): along-slope component of gravity = $mg\sin\theta$; perpendicular component = $mg\cos\theta$. Normal force = $mg\cos\theta$. Friction along slope = $\mu mg\cos\theta$.

Tension in a massless, inextensible rope is the same throughout. On a frictionless pulley, tension is equal on both sides.

Centripetal (center-seeking) force - any force holding an object in circular motion:
$F_c = \frac{mv^2}{r} = m\omega^2 r$
$a_c = \frac{v^2}{r} = \omega^2 r$

Centripetal acceleration points toward the center. “Centrifugal force” is a fictitious force in a rotating reference frame - the MCAT wants centripetal.

Universal gravitation: $F = \dfrac{G m_1 m_2}{r^2}$, $G = 6.67 \times 10^{-11}~\mathrm{N\,m^2/kg^2}$. Surface gravity: $g = GM/R^2$. Orbital speed: $v = \sqrt{GM/r}$.

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4. Work, Energy, and Power

Work done by a constant force on a displacement:
$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$

θ is the angle between force and displacement. Force perpendicular to motion does zero work (normal force on flat surface, centripetal force, magnetic force on moving charge).

Kinetic energy: $KE = \tfrac{1}{2}mv^2$. Always positive.

Gravitational potential energy (near Earth): $PE_g = mgh$. Reference point is arbitrary - only changes in PE matter.

Elastic (spring) potential energy: $PE_s = \tfrac{1}{2}kx^2$, where $k$ = spring constant, $x$ = displacement from equilibrium. Hooke’s Law: $F = -kx$.

Power: $P = W/t = \vec{F}\cdot\vec{v} = Fv\cos\theta$. Unit: watt (1 W = 1 J/s). 1 horsepower ≈ 746 W.

Efficiency: $\eta = W_{\text{useful}} / E_{\text{input}}$. Always ≤ 1.

Simple machines (lever, pulley, inclined plane) trade force for distance - they multiply force while increasing the distance over which you apply it. Ideal mechanical advantage × ideal distance ratio = 1. Real efficiency < 1 due to friction.

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5. Momentum, Impulse, and Collisions

Momentum: $\vec{p} = m\vec{v}$. Unit: kg·m/s. Always conserved in an isolated system (no external forces).

Impulse-momentum theorem:
$\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p} = m\Delta \vec{v}$

Airbags, crumple zones, follow-through in sports: all increase Δt to reduce peak force for the same momentum change.

Collision Type	KE conserved?	Momentum conserved?	Example
Elastic	Yes	Yes	Billiard balls, gas molecules
Inelastic	No (some → heat/sound)	Yes	Most real collisions
Perfectly inelastic	No (maximum KE lost)	Yes	Objects stick together

Perfectly inelastic collision (objects stick): $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$.

1-D elastic collision between $m_1$ (moving at $v_1$) and $m_2$ (at rest):
$v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_1, \qquad v_{2f} = \frac{2 m_1}{m_1 + m_2} v_1$

Equal masses in 1-D elastic: $m_1$ stops, $m_2$ takes all velocity (classic pool-ball collision).

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6. Rotational Motion, Torque, and Center of Mass

Angular quantities: $\theta$ (rad), $\omega$ (rad/s), $\alpha$ (rad/s²). Conversions: $s = r\theta$, $v = r\omega$, $a_t = r\alpha$. One full revolution = 2π rad = 360°.

Torque: $\tau = rF\sin\theta = r_\perp F$. Unit: N·m. Torque is maximized when force is perpendicular to the lever arm (θ = 90°).

Rotational equilibrium: $\sum \tau = 0$ (and $\sum F = 0$ for translational equilibrium). Used for see-saws, bridges, static limb mechanics.

Rotational analogs:

Linear	Rotational
$F = ma$	$\tau = I\alpha$
$KE = \tfrac{1}{2}mv^2$	$KE_{rot} = \tfrac{1}{2}I\omega^2$
$p = mv$	$L = I\omega$

Angular momentum $L = I\omega$ is conserved when net external torque is zero (skater pulling arms in spins faster).

Center of mass for a system of particles:
$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

External forces accelerate the center of mass; internal forces do not change it.

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7. Fluids

Density: $\rho = m/V$. Specific gravity: $SG = \rho/\rho_{water}$. Water has $\rho = 1000~\mathrm{kg/m^3}$ = 1 g/cm³.

Pressure: $P = F/A$. Unit: pascal (1 Pa = 1 N/m²). 1 atm = 1.013 × 10⁵ Pa = 760 mmHg = 760 torr.

Hydrostatic (fluid column) pressure: pressure at depth $h$ below the surface of a static fluid:
$P = P_0 + \rho g h$

Pressure depends only on depth, not shape of container.

Poiseuille’s law (laminar flow, viscous fluid, cylindrical tube):
$Q = \frac{\pi r^4 \Delta P}{8 \eta L}$

Flow rate scales with the fourth power of radius - a 2× wider vessel carries 16× more flow. Key to how vasoconstriction spikes blood pressure.

Reynolds number: $Re = \rho v D / \eta$. $Re < 2000$ → laminar; $Re > 3000$ → turbulent.

Viscosity $\eta$: internal friction in a fluid (high in honey, low in water). Increases with lower temperature in liquids.

Surface tension: cohesive pull at a fluid surface - lets water striders float, produces capillary action.

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8. Thermodynamics

Temperature scales: $T_K = T_C + 273.15$. $T_F = \tfrac{9}{5} T_C + 32$. Absolute zero = 0 K = -273.15 °C.

Heat transfer modes: conduction (direct contact), convection (bulk fluid motion), radiation (EM waves, no medium needed). Radiation obeys Stefan-Boltzmann $P = e\sigma A T^4$.

Specific heat: $Q = mc\Delta T$. Water has unusually high $c = 4186~\mathrm{J/(kg\cdot K)}$.

Latent heat (phase change at constant T): $Q = mL$. $L_f$ (fusion), $L_v$ (vaporization). No temperature change during a phase transition - all heat goes into breaking intermolecular forces.

Thermal expansion: linear $\Delta L = \alpha L_0 \Delta T$; volumetric $\Delta V = \beta V_0 \Delta T$, with $\beta \approx 3\alpha$.

Ideal gas law: $PV = nRT$. $R = 8.314~\mathrm{J/(mol\cdot K)} = 0.0821~\mathrm{L\cdot atm/(mol\cdot K)}$. At STP, 1 mol = 22.4 L.

Laws of thermodynamics:

• 0th: if $A$ and $B$ are each in thermal equilibrium with $C$, then $A$ and $B$ are in equilibrium with each other (defines temperature).
• 1st: $\Delta U = Q - W$. Internal energy change = heat added to system minus work done by the system. (Physics sign convention - chem often uses $\Delta U = Q + W$ where $W$ is work done on the system.)
• 2nd: entropy of an isolated system never decreases. Heat flows spontaneously from hot to cold.
• 3rd: entropy of a perfect crystal at 0 K is zero.

Process	Constant	Formula
Isothermal	T	$\Delta U = 0$; $Q = W = nRT\ln(V_f/V_i)$
Adiabatic	Q (no heat exchange)	$Q = 0$; $\Delta U = -W$
Isobaric	P	$W = P\Delta V$; $Q = nC_p\Delta T$
Isochoric (isovolumetric)	V	$W = 0$; $\Delta U = Q = nC_v\Delta T$

Efficiency of a heat engine: $\eta = W/Q_H = 1 - Q_C/Q_H$. Carnot (max) efficiency: $\eta_C = 1 - T_C/T_H$ (in Kelvin).

Entropy change at constant T: $\Delta S = Q_{rev}/T$.

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9. Waves and Sound

Wave equation: $v = f\lambda$. Speed depends on the medium, not the source.

Period and frequency: $T = 1/f$. Angular frequency $\omega = 2\pi f$.

Wave types: transverse (oscillation ⊥ propagation - light, string) vs longitudinal (oscillation ∥ propagation - sound, compression waves).

Superposition: waves add algebraically. Constructive: in phase (Δφ = 0, 2π, 4π…). Destructive: out of phase (Δφ = π, 3π…).

Speed of sound in air: ~343 m/s at 20°C (faster in warmer air, faster in denser/stiffer solids).

Doppler effect:
$f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$

• Source/observer approach: frequency heard ↑ (top sign pair)
• Source/observer recede: frequency heard ↓ (bottom sign pair)

Use the sign that makes the numerator larger (approaching) or denominator larger (receding).

Standing waves on a string (both ends fixed): $\lambda_n = 2L/n$, $f_n = nv/(2L)$, $n = 1, 2, 3, \dots$ (“harmonics” - whole-number multiples).

Pipes (air columns):

Pipe	Allowed harmonics	Formula
Both ends open	$n = 1, 2, 3, \dots$	$f_n = nv/(2L)$
One end closed	$n = 1, 3, 5, \dots$ (odd only)	$f_n = nv/(4L)$

Fundamental (1st harmonic, $n=1$) is the lowest frequency. Closed pipes have a node at the closed end, antinode at the open end.

Beats: two close frequencies produce a beat frequency equal to $|f_1 - f_2|$. Used to tune instruments.

Sound intensity $I = P/A$, measured in W/m². Decibel level:
$\beta = 10 \log\frac{I}{I_0}, \quad I_0 = 10^{-12}~\mathrm{W/m^2}$

Each 10 dB increase = 10× intensity. Each 20 dB ≈ 2× perceived loudness.

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10. Light and Optics

Light is an electromagnetic wave. $c = 3 \times 10^8$ m/s in vacuum. Energy per photon: $E = hf = hc/\lambda$, with $h = 6.63 \times 10^{-34}$ J·s.

EM spectrum (low to high frequency): radio → microwave → infrared → visible (700-400 nm, red to violet - “ROY G BIV”) → UV → X-ray → gamma.

Reflection: angle of incidence = angle of reflection (both measured from the normal).

Refraction (Snell’s law): $n_1 \sin\theta_1 = n_2 \sin\theta_2$. Index of refraction $n = c/v$ (always ≥ 1). Light bends toward the normal when entering a denser medium.

Total internal reflection occurs when going from denser → less dense medium above the critical angle:
$\sin\theta_c = n_2/n_1 \quad (\text{for } n_1 > n_2)$

Basis of fiber optics.

Sign conventions (memorize cold):

Quantity	Positive	Negative
$f$ (mirror)	Concave (converging)	Convex (diverging)
$f$ (lens)	Convex (converging)	Concave (diverging)
$d_i$	Real image (same side as outgoing light)	Virtual image (opposite side)
$m$	Upright image	Inverted image
$	m	> 1$
$	m	< 1$

Power of a lens: $P = 1/f$ (in meters), unit: diopter. Two lenses in contact: $P_{\text{total}} = P_1 + P_2$.

Polarization: transverse waves only. Unpolarized → polarized: intensity drops by half. Malus’s law (polarizer to polarizer): $I = I_0 \cos^2\theta$.

Diffraction and interference:

• Double slit (maxima): $d\sin\theta = m\lambda$, $m = 0, 1, 2, \dots$
• Single slit (minima, dark fringes): $a\sin\theta = m\lambda$, $m = 1, 2, 3, \dots$

Pattern is brightest at the center (m = 0) with fringes spaced farther apart for smaller $d$ (slit separation) or longer $\lambda$.

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11. Electrostatics

Coulomb’s law:
$F = \frac{k q_1 q_2}{r^2}, \quad k = 8.99 \times 10^9~\mathrm{N\,m^2/C^2}$

Like charges repel, opposites attract. Inverse-square law (same form as gravity, much stronger).

Electric field (force per unit positive charge): $\vec{E} = \vec{F}/q$. From a point charge: $E = kQ/r^2$. Field lines point away from + and toward -.

Electric potential (voltage) from a point charge: $V = kQ/r$. Between plates: $V = Ed$. Potential energy:
$U = \frac{kq_1 q_2}{r}, \qquad U = qV$

Work by an electric force: $W = q\Delta V$. Electron-volt: 1 eV = 1.6 × 10⁻¹⁹ J = energy gained by an electron moving through 1 V.

Capacitors: $C = Q/V$. Unit: farad (F). Parallel-plate: $C = \varepsilon_0 A/d$; with dielectric: $C = \kappa \varepsilon_0 A/d$ (κ ≥ 1).

Energy stored in a capacitor:
$U = \tfrac{1}{2} CV^2 = \tfrac{1}{2} QV = \tfrac{Q^2}{2C}$

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12. Circuits

Ohm’s law: $V = IR$. Current $I$ (amperes) = charge per time = $Q/t$.

Resistance: $R = \rho L/A$ (ρ = resistivity). Longer or thinner wire → more resistance.

Power:
$P = IV = I^2 R = \frac{V^2}{R}$

Units: watts. Electric bill = energy = power × time (kWh).

Series circuit (one loop):

• Same current through every element
• Voltage splits: $V_{total} = V_1 + V_2 + \dots$
• $R_{eq} = R_1 + R_2 + \dots$
• Capacitors: $\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dots$

Parallel circuit (branches):

• Same voltage across every branch
• Current splits: $I_{total} = I_1 + I_2 + \dots$
• $\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots$
• Capacitors: $C_{eq} = C_1 + C_2 + \dots$

Kirchhoff’s laws:

• Junction (current) rule: $\sum I_{in} = \sum I_{out}$ (charge conservation).
• Loop (voltage) rule: sum of voltage drops around any closed loop = 0 (energy conservation).

RC circuit (charging through resistor):
$Q(t) = Q_{max}(1 - e^{-t/\tau}), \qquad \tau = RC$

Time constant $\tau$ = time to reach ~63% of final value. Discharging: $Q(t) = Q_0 e^{-t/\tau}$.

Batteries and emf: terminal voltage $V = \varepsilon - Ir$ (emf minus drop across internal resistance).

AC vs DC: DC = constant direction; AC = sinusoidal, $V(t) = V_0 \sin(\omega t)$. $V_{rms} = V_0/\sqrt{2}$, $I_{rms} = I_0/\sqrt{2}$. Household AC: $V_{rms} = 120$ V, so $V_0 \approx 170$ V.

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13. Magnetism

Magnetic force on a moving charge:
$\vec{F} = q\vec{v} \times \vec{B}, \qquad |F| = qvB\sin\theta$

Force is perpendicular to both $\vec{v}$ and $\vec{B}$. Does zero work on the charge (only changes direction). Charges moving ⊥ to a uniform B execute circular motion with $r = mv/(qB)$.

Force on a current-carrying wire:
$\vec{F} = I\vec{L} \times \vec{B}, \qquad |F| = BIL\sin\theta$

Right-hand rule (for positive charges / conventional current): point fingers in direction of $\vec{v}$ or $\vec{I}$, curl toward $\vec{B}$, thumb gives $\vec{F}$. For electrons/negative charges, reverse direction.

Magnetic field from a long straight wire: $B = \mu_0 I / (2\pi r)$.

Solenoid: $B = \mu_0 n I$ (n = turns per length). Inside a solenoid, B is uniform and parallel to the axis.

Faraday’s law (induction): changing magnetic flux induces an emf. $\varepsilon = -d\Phi_B/dt$. Lenz’s law (sign): induced current opposes the change in flux that caused it.

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14. Atomic and Nuclear Physics

Photoelectric effect: light of frequency $f$ ejects electrons from a metal only if $f \ge f_0$ (threshold). Kinetic energy of ejected electron:
$KE_{max} = hf - \phi$

$\phi$ = work function of the metal. Evidence that light behaves as particles (photons).

Bohr model: energy levels in hydrogen: $E_n = -13.6/n^2$ eV. Transition from $n_i$ to $n_f$ emits/absorbs a photon of $\Delta E = E_{n_i} - E_{n_f}$. Balmer series = visible (transitions to n=2).

Mass-energy equivalence: $E = mc^2$. Nuclear reactions involve mass defects convertible to enormous energy.

Nuclear notation: $^A_Z X$ where $A$ = mass number (protons + neutrons), $Z$ = atomic number (protons).

Decay	Emits	$\Delta Z$	$\Delta A$
Alpha (α)	$^4_2 \text{He}$	$-2$	$-4$
Beta minus (β⁻)	$e^-$ + antineutrino	$+1$	$0$
Beta plus (β⁺)	$e^+$ + neutrino	$-1$	$0$
Gamma (γ)	high-energy photon	$0$	$0$
Electron capture	neutrino	$-1$	$0$

Half-life: $N(t) = N_0 (1/2)^{t/t_{1/2}}$. After $n$ half-lives, $1/2^n$ remains. Exponential form: $N(t) = N_0 e^{-\lambda t}$, $t_{1/2} = \ln 2 / \lambda \approx 0.693/\lambda$.

Binding energy per nucleon: measure of nuclear stability. Peaks near iron-56. Fusion (light nuclei) and fission (heavy nuclei) both release energy by moving toward iron.

Mass defect: difference between the mass of separated nucleons and the mass of the bound nucleus; $\Delta m \cdot c^2$ = binding energy.

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15. Statistics and Research Methods

Central tendency:

• Mean: sum / count. Sensitive to outliers.
• Median: middle value. Robust to outliers.
• Mode: most frequent value.

For a skewed distribution: right-skewed (long right tail) → mean > median > mode; left-skewed → mean < median < mode.

Spread:

• Range: max - min.
• Variance $\sigma^2 = \sum (x_i - \bar{x})^2 / N$.
• Standard deviation $\sigma = \sqrt{\sigma^2}$.
• Standard error of the mean: $SEM = \sigma/\sqrt{n}$.

Normal distribution (68-95-99.7 rule): ~68% of data within 1σ of the mean, ~95% within 2σ, ~99.7% within 3σ.

Correlation vs causation: a correlation coefficient $r$ runs from -1 (perfect negative) to +1 (perfect positive). Correlation alone does not prove causation.

Study designs:

Type	Description	Strength
Case study	Single patient/event	Hypothesis-generating only
Cross-sectional	Snapshot of a population	Fast; cannot show causation
Case-control	Compare people with/without outcome, look backward	Good for rare diseases; recall bias
Cohort	Follow a group over time	Shows incidence; expensive
RCT	Randomized, controlled, blinded	Gold standard for causation
Meta-analysis	Pool many studies	Highest level of evidence

Variables: independent (manipulated), dependent (measured), confounding (correlates with both), control (held constant).

Validity vs reliability:

• Internal validity: study measures what it claims within itself.
• External validity: results generalize beyond the sample.
• Reliability: repeatable results (test-retest, inter-rater).

A test can be reliable without being valid (consistently wrong), but not valid without being reliable.

P-value: probability of observing data as extreme as yours if the null hypothesis is true. $p < 0.05$ = typically “statistically significant” (arbitrary threshold).

• Type I error (α): false positive - rejecting a true null.
• Type II error (β): false negative - failing to reject a false null.
• Power = 1 - β = probability of correctly detecting a real effect.

Confidence interval: a 95% CI means 95% of such intervals (computed this way across many samples) would contain the true population parameter. If a 95% CI for a difference contains 0, the result is not significant at α = 0.05.

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16. The Master Formula Sheet

Single-source reference - this is what to put on a flashcard per section.

Kinematics: $v = v_0 + at$; $x = v_0 t + \tfrac{1}{2}at^2$; $v^2 = v_0^2 + 2a\Delta x$.

Dynamics: $F = ma$; $W = mg$; $f = \mu N$; $F_c = mv^2/r$; $F_g = Gm_1m_2/r^2$.

Energy: $W = Fd\cos\theta$; $KE = \tfrac{1}{2}mv^2$; $PE_g = mgh$; $PE_s = \tfrac{1}{2}kx^2$; $P = W/t$.

Momentum: $p = mv$; $J = F\Delta t = \Delta p$.

Rotation: $\tau = rF\sin\theta$; $L = I\omega$; $KE_{rot} = \tfrac{1}{2}I\omega^2$.

Fluids: $\rho = m/V$; $P = F/A$; $P = P_0 + \rho gh$; $F_B = \rho_f V g$; $A_1 v_1 = A_2 v_2$; $P + \tfrac{1}{2}\rho v^2 + \rho gh = \text{const}$; $Q = \pi r^4 \Delta P/(8\eta L)$.

Thermo: $Q = mc\Delta T$; $Q = mL$; $PV = nRT$; $\Delta U = Q - W$; $\eta_C = 1 - T_C/T_H$.

Waves: $v = f\lambda$; $f' = f(v\pm v_o)/(v\mp v_s)$; $f_n = nv/(2L)$ open/string; $f_n = nv/(4L)$ closed; $\beta = 10\log(I/I_0)$.

Optics: $n_1\sin\theta_1 = n_2\sin\theta_2$; $1/f = 1/d_o + 1/d_i$; $m = -d_i/d_o$; $d\sin\theta = m\lambda$ (double slit max); $I = I_0 \cos^2\theta$.

Electrostatics: $F = kq_1 q_2/r^2$; $E = kQ/r^2$; $V = kQ/r$; $U = qV$; $C = Q/V$; $U = \tfrac{1}{2}CV^2$.

Circuits: $V = IR$; $P = IV$; $R_s = \sum R_i$; $1/R_p = \sum 1/R_i$; $\tau = RC$.

Magnetism: $F = qvB\sin\theta$; $F = BIL\sin\theta$; $B = \mu_0 I/(2\pi r)$.

Atomic/Nuclear: $E = hf = hc/\lambda$; $KE_{max} = hf - \phi$; $E_n = -13.6/n^2$ eV; $N(t) = N_0 (1/2)^{t/t_{1/2}}$; $E = mc^2$.

Constants to memorize: $g = 9.8~\mathrm{m/s^2}$; $c = 3 \times 10^8$ m/s; $k = 9 \times 10^9$ N·m²/C²; $e = 1.6 \times 10^{-19}$ C; $h = 6.63 \times 10^{-34}$ J·s; $R = 8.314$ J/(mol·K); $N_A = 6.02 \times 10^{23}$ /mol; 1 atm = 10⁵ Pa; $\rho_{water} = 1000$ kg/m³.

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17. Top 25 Facts to Memorize

1. Projectile max range at 45° (equal launch/land height); horizontal velocity constant throughout.
2. At the top of a projectile’s path, $v_y = 0$ but $v_x$ is unchanged.
3. Newton’s 2nd: $F_{\text{net}} = ma$. Always draw a free-body diagram first.
4. Friction $f = \mu N$; normal force on incline = $mg\cos\theta$.
5. Centripetal force points toward the center: $F_c = mv^2/r$.
6. Work by perpendicular force = 0 (normal force, centripetal, magnetic force on moving charge).
7. Conservation of energy: $KE + PE = \text{const}$ with only conservative forces.
8. Power = Fv (when F ∥ v); 1 W = 1 J/s.
9. Momentum is always conserved in isolated collisions; KE only in elastic.
10. Equal-mass 1-D elastic collision: striker stops, target takes all velocity.
11. Pressure at depth: $P = P_0 + \rho gh$ - depends on depth only, not shape.
12. Buoyant force = weight of fluid displaced. Float if $\rho_{obj} < \rho_{fluid}$.
13. Continuity: $A_1 v_1 = A_2 v_2$. Narrow → fast.
14. Bernoulli: fast fluid = low pressure (at the same height).
15. Poiseuille: $Q \propto r^4$ - radius is the blockbuster variable.
16. Ideal gas law: $PV = nRT$. At STP, 1 mol = 22.4 L.
17. 1st law of thermo: $\Delta U = Q - W$ (heat in, minus work done by system).
18. Adiabatic: $Q = 0$. Isothermal: $\Delta U = 0$.
19. Wave speed depends on the medium, not the source. $v = f\lambda$.
20. Doppler: approaching → higher frequency; receding → lower.
21. Standing wave pipes: open-open or both-fixed string → all harmonics; closed-open pipe → odd harmonics only.
22. Snell’s law: $n_1\sin\theta_1 = n_2\sin\theta_2$. Bends toward normal in denser medium.
23. Thin lens/mirror: $1/f = 1/d_o + 1/d_i$; $m = -d_i/d_o$. Real image → $d_i > 0$, inverted.
24. Ohm’s law: $V = IR$. Series: R adds, C reciprocal. Parallel: R reciprocal, C adds.
25. Half-life: after $n$ half-lives, $1/2^n$ remains. Alpha drops Z by 2 and A by 4; beta⁻ raises Z by 1, A unchanged.

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18. Test-Day Mnemonics

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Next Steps

If something on this page feels unfamiliar, open the corresponding chapter and re-derive the formula from first principles. Physics is the section where formulas are the vocabulary and the free-body diagram is the grammar. This cheat sheet gets you to 80% recall on test day; the full chapters get you the reasoning that lets you solve unfamiliar passages under the clock. Bookmark this page and return to it the night before every practice test and on the morning of the real thing.