Physics Formulas Cheat Sheet by Topic: Mechanics, E&M, Waves, Thermodynamics, and Modern Physics

Overview

This article is a reusable physics formulas cheat sheet by topic. It is not meant to replace a full set of physics notes. Instead, it acts as a compact reference you can return to whenever you need to refresh the structure of a chapter quickly.

A useful formula sheet should answer four questions:

• What does the equation calculate?
• When is it valid?
• What units should the variables use?
• What nearby equation is easy to confuse with it?

That last point matters. Many students do not struggle because they have never seen the equation. They struggle because several equations look related and only one matches the assumptions in the problem. If you want a broader study system around quick review and retention, see AI Study Guides and the Physics of Learning: What Makes Notes Turn into Knowledge?.

Before the topic-by-topic checklist, keep these general rules in view:

• Use SI units unless a problem clearly asks otherwise.
• Write known quantities first, with units.
• Sketch the situation if motion, fields, or geometry are involved.
• Check whether the formula is scalar or vector in meaning.
• After solving, ask whether the size and sign of your answer make physical sense.

Below, the formulas are grouped by scenario rather than by textbook chapter title alone. That makes the sheet easier to use under exam pressure.

Checklist by scenario

Use this section as the main reference. Each group includes staple equations, when to use them, and what to watch for.

1) Mechanics formulas: motion, forces, energy, and momentum

Start here when a problem involves objects moving, speeding up, colliding, rotating, or staying in equilibrium.

Kinematics for constant acceleration

• v = v₀ + at — use when acceleration is constant and you want final velocity after time t.
• x = x₀ + v₀t + (1/2)at² — use for position change under constant acceleration.
• v² = v₀² + 2a(x - x₀) — useful when time is not given.

Double-check: these equations do not apply if acceleration changes significantly with time unless the problem says to treat it as constant.

Newton's laws and force relations

• F = ma — the core relation between net force and acceleration.
• Weight: W = mg — force due to gravity near Earth's surface.
• Friction: f_s ≤ μ_sN, f_k = μ_kN — distinguish static from kinetic friction.
• Hooke's law: F = -kx — restoring force of an ideal spring.

Common use case: free-body diagrams. Write all forces first, then apply Newton's second law along chosen axes.

Work and energy

• Work: W = Fd cosθ — for constant force over displacement.
• Kinetic energy: K = (1/2)mv²
• Gravitational potential energy: U = mgh — near Earth's surface.
• Spring potential energy: U = (1/2)kx²
• Power: P = W/t = Fv for motion parallel to force.

Use energy methods when force directions change, or when a problem asks for speed, height, or compression without requiring detailed time information.

Momentum and impulse

• Momentum: p = mv
• Impulse: J = FΔt = Δp
• Conservation of momentum: total momentum before = total momentum after in an isolated system.

Best for: collisions, explosions, recoil, and short-duration interactions.

Circular motion and gravitation

• Centripetal acceleration: a_c = v²/r
• Centripetal force: F_c = mv²/r
• Universal gravitation: F = Gm₁m₂/r²

Do not treat centripetal force as a new physical force. It is the net inward force supplied by tension, gravity, friction, or some combination of forces.

If you enjoy applied examples, orbital motion problems connect naturally with Apollo 13, Artemis II, and the Physics of Going Around the Moon.

2) Electromagnetism formulas: charge, fields, circuits, and magnetic effects

Use these formulas when charges interact, electric fields do work, circuits carry current, or moving charges feel magnetic forces.

Electrostatics

• Coulomb's law: F = kq₁q₂/r²
• Electric field: E = F/q
• Field of a point charge: E = kq/r²
• Electric potential energy: U = qV
• Potential difference: V = W/q

Key distinction: field is force per charge, while potential is energy per charge. These are closely related but not interchangeable.

Basic circuit formulas

• Current: I = ΔQ/Δt
• Ohm's law: V = IR
• Electrical power: P = IV = I²R = V²/R
• Equivalent resistance in series: R_eq = R₁ + R₂ + ...
• Equivalent resistance in parallel: 1/R_eq = 1/R₁ + 1/R₂ + ...

Use case: start with a clean circuit redraw. Then simplify series and parallel sections step by step before calculating currents or voltages.

Capacitance and stored energy

• Capacitance: C = Q/V
• Energy stored in a capacitor: U = (1/2)CV²

Magnetism

• Magnetic force on a charge: F = qvB sinθ
• Magnetic force on a current-carrying wire: F = ILB sinθ
• Magnetic flux: Φ_B = BA cosθ
• Faraday's law, magnitude form: |ε| = |ΔΦ_B/Δt|

Watch the angle. In magnetic force problems, the force depends on the angle between velocity and field. In flux problems, it depends on the angle between the magnetic field and the surface normal.

3) Waves and oscillations formulas: frequency, speed, sound, and light

Use this set when a problem involves repeating motion, traveling disturbances, resonance, or optics basics.

Wave basics

• Wave speed: v = fλ
• Frequency-period relation: f = 1/T
• Angular frequency: ω = 2πf

Simple harmonic motion

• Spring-mass period: T = 2π√(m/k)
• Pendulum period for small angles: T = 2π√(L/g)

Sound and standing waves

• Fundamental string frequency: f = v/(2L) for a string fixed at both ends.
• Open pipe fundamental: f = v/(2L)
• Closed pipe fundamental: f = v/(4L)

Optics basics

• Index of refraction: n = c/v
• Snell's law: n₁ sinθ₁ = n₂ sinθ₂
• Thin lens / mirror equation: 1/f = 1/d_o + 1/d_i
• Magnification: m = -d_i/d_o = h_i/h_o

Most common confusion: the symbol f can mean frequency or focal length depending on the topic. Read the context before substituting numbers.

4) Thermodynamics equations: heat, ideal gases, and energy flow

Turn to these formulas when temperature changes, gases compress or expand, or a problem asks about heat and work.

Temperature and heat

• Heat transfer: Q = mcΔT
• Phase change: Q = mL

Use case: calorimetry, heating curves, and mixing problems.

Ideal gas relations

• Ideal gas law: PV = nRT
• Combined gas law forms can be derived when amount of gas is constant.

Thermodynamic bookkeeping

• First law of thermodynamics: ΔU = Q - W using the common sign convention where W is work done by the system.

Important habit: write your sign convention on the page. Some courses define work with the opposite sign. Many mistakes in thermodynamics are not conceptual; they are bookkeeping errors.

Efficiency and engines

• Efficiency: e = W_out/Q_in

Even if your course keeps thermodynamics introductory, pay attention to units: joules for energy, kelvin for absolute temperature, pascals for pressure, and cubic meters for volume.

5) Modern physics formulas: photons, atoms, relativity, and quantum basics

This section is often shorter in an introductory class, but the equations are conceptually dense, so a cheat sheet is especially helpful.

Photon and wave-particle formulas

• Photon energy: E = hf
• Wavelength relation: c = fλ
• de Broglie wavelength: λ = h/p

Atomic and nuclear basics

• Mass-energy equivalence: E = mc²

Relativity basics

• Lorentz factor: γ = 1/√(1 - v²/c²)

Use with care: relativistic formulas matter when speeds are a significant fraction of the speed of light. For ordinary classroom mechanics problems, classical formulas are usually enough unless the problem signals otherwise.

Quantum notation reminder

In modern physics, symbols may carry meanings that are less intuitive than in mechanics. A quick note beside each symbol can prevent confusion. If you want a conceptual bridge into quantum computing language, Why Quantum Computers Need Standards: What a Logical Qubit Actually Is is a useful companion read.

What to double-check

Before finalizing any solution, run through this short checklist. It catches a surprising number of errors.

• Units: Did you convert centimeters to meters, grams to kilograms, degrees Celsius to kelvin where needed, and hours to seconds?
• Symbols: Does the letter mean what you think it means in this chapter? For example, f may mean frequency or focal length.
• Angles: Is the angle in the formula measured from the correct direction? This is crucial for work, magnetic force, and flux.
• System boundaries: In momentum or energy problems, what exactly is included in the system?
• Sign convention: Are you treating direction, electric charge sign, and thermodynamic work consistently?
• Assumptions: Is acceleration constant? Is friction negligible? Is the process idealized?
• Reasonableness: Should the answer be positive, less than one, larger than the starting value, or physically impossible under the conditions?

It also helps to compare formulas that are easy to mix up. For example:

• F = ma versus F = mv²/r: the second is a special case of net inward force in circular motion.
• Q = mcΔT versus Q = mL: the first changes temperature, the second changes phase.
• E = F/q versus V = W/q: field and potential are different quantities.
• p = mv versus K = (1/2)mv²: momentum and kinetic energy respond differently to changes in velocity.

For students trying to become more careful readers of technical claims and textbook shortcuts, How to Spot a Physics Textbook Claim: A Fact-Checking Toolkit for Students offers a useful mindset.

Common mistakes

Most formula errors are not about forgetting everything. They come from a small set of repeated habits.

• Plugging numbers in too early. Keep symbols until the setup is correct. This makes rearrangement easier and reveals unit problems sooner.
• Using memorized formulas without checking conditions. A constant-acceleration equation is not a universal motion equation.
• Ignoring vectors. Adding speeds, forces, or fields without direction leads to wrong magnitudes and signs.
• Dropping units in intermediate steps. Units are part of the calculation, not decoration at the end.
• Confusing net force with a single force. In circular motion and equilibrium problems, always identify all contributing forces.
• Mixing chapter language. The same symbol can shift meaning across mechanics, waves, optics, and thermodynamics.
• Forgetting limiting cases. If friction is zero, if angle is 90 degrees, if temperature change is zero—your formula should reflect that cleanly.

A practical fix is to build your own two-column reference: formula on the left, “when I use this” on the right. That turns a static physics equations list into a working study aid.

When to revisit

This cheat sheet is most useful when you return to it regularly rather than only the night before an exam. Revisit and update your formula list in these situations:

• At the start of a new unit: add the 5 to 10 equations that define the chapter.
• Before quizzes and exams: mark which formulas are often paired together in multi-step problems.
• After graded work is returned: note which mistakes came from setup, units, sign conventions, or equation choice.
• When your course changes level: AP Physics, IB Physics, algebra-based college physics, and calculus-based physics often emphasize different forms and assumptions.
• When your workflow changes: if you switch from paper notes to digital flashcards or simulation-based study, revise the sheet so it stays usable.

Here is a simple action plan you can use today:

1. Create one page per topic: mechanics, E&M, waves, thermodynamics, modern physics.
2. For every formula, add three labels: meaning, units, and use case.
3. Highlight formulas you confuse with similar ones.
4. After each problem set, add one note about a mistake you want to avoid next time.
5. Review the sheet briefly every week instead of cramming it all at once.

If you treat your formula sheet as a living document, it becomes more than an exam tool. It becomes a compact map of physics explained in the language of quantities, patterns, and constraints. That is exactly why a well-built reference stays useful semester after semester.