The problem: the formula is right, the answer is wrong
You pick the right formula, plug in the numbers, and get an answer that's off by a factor of 1,000. The formula wasn't wrong. The units were. You put distance in kilometers when the formula expects meters, or temperature in Celsius when it needs Kelvin, and the equation happily multiplied them and gave you a number that looks plausible but is wrong by three orders of magnitude. Physics formulas don't check units. They multiply whatever you give them. The formula is a relationship between quantities, and if the quantities are in the wrong units, the relationship breaks.
This is the single most common error in physics problem-solving, and it's not a formula problem — it's a discipline problem. The formula reference gives you the equation and the units each variable expects. Your job is to convert your inputs to those units before plugging them in.
Fastest path
Open the Physics Formulas tool, search or browse by topic (Mechanics, Waves & Sound, Optics, Thermodynamics, Electricity, Modern Physics), and click any formula to see its variables, units, and description. For most formulas, the built-in solver lets you enter known values and solve for any unknown variable — no algebraic rearrangement required.
Worked example — a 2 kg object accelerates at 3 m/s². What force is applied?
Formula: F = m × a
Inputs: m = 2 kg, a = 3 m/s²
Result: F = 6 N
The tool handles the algebra. You handle the units.
The five formula families
Physics formulas fall into five families, and knowing which family you're in tells you which formula to reach for.
Kinematics: describing motion (no forces)
Kinematics formulas describe how things move without asking why. They relate position, velocity, acceleration, and time. The tool includes the two most-used equations:
v = u + a·t (final velocity from initial velocity, acceleration, time)
s = u·t + ½·a·t² (displacement from initial velocity, acceleration, time)
Two more that complete the set (not in the tool but worth knowing):
v² = u² + 2·a·s (velocity from displacement, no time variable)
s = ½·(u + v)·t (displacement from average velocity, no acceleration variable)
The trick is picking the equation that uses the variables you know and the one you want. If you know time, use the first two. If you don't know time but know displacement, use the third. If you don't know acceleration, use the fourth.
Dynamics: forces cause motion
Dynamics formulas explain why motion happens. Newton's second law is the centerpiece:
F = m × a
If you know the force and mass, you get acceleration. If you know the force and acceleration, you get mass. The tool's solver does this rearrangement automatically. Related formulas in this family: weight (W = m·g, where g is gravitational acceleration), friction (f = μ·N, where μ is the coefficient of friction and N is normal force), and Newton's law of gravitation (F = G·m₁·m₂/r²).
Energy and work: scalar conservation
Energy formulas track a scalar quantity that's conserved (it changes form but doesn't disappear). The three you'll use most:
KE = ½·m·v² (kinetic energy — energy of motion)
PE = m·g·h (gravitational potential energy — energy of position)
W = F·d·cos(θ) (work — energy transferred by a force over a distance)
The angle θ in the work formula matters. If you push a box horizontally (θ = 0°, cos(0°) = 1), all your force does work. If you push straight down on the same box (θ = 90°, cos(90°) = 0), you do zero work on the horizontal motion. The tool includes θ as a variable, and it's the one people forget most often.
Momentum: vector conservation
Momentum is a vector (it has direction), and it's conserved in collisions and explosions:
p = m × v (momentum = mass × velocity)
Conservation of momentum means the total momentum before a collision equals the total after. This is how you analyze car crashes, billiard ball collisions, and rocket propulsion — the rocket goes forward because exhaust goes backward, and the total momentum stays zero.
Fields: the inverse square laws
Two of the most important formulas in physics share the same structure — force proportional to 1/r²:
F = G·m₁·m₂ / r² (gravity, G = 6.674 × 10⁻¹¹ N·m²/kg²)
F = k·q₁·q₂ / r² (electrostatic, k = 8.99 × 10⁹ N·m²/C²)
The inverse square means doubling the distance reduces the force by a factor of 4, not 2. Tripling the distance reduces it by 9. This is why gravity feels strong on Earth but negligible at the Moon's orbit, and why static electricity zaps you when you're close to a charged surface but not when you're a meter away.
The constants that aren't constant
The tool pre-loads several physical constants. Most are truly constant. Two are not:
g (gravitational acceleration, 9.81 m/s²). This is the value at Earth's surface, at sea level, at mid-latitudes. At the top of Everest it's about 9.77. On the Moon it's 1.62. On Mars it's 3.71. If you're calculating weight on another planet, change g. The tool's weight formula (W = m·g) lets you override g — enter the local value and the calculation is correct for that location.
R (gas constant, 8.314 J/(mol·K)). This one is truly constant — it's a fundamental physical constant. But the ideal gas law (PV = nRT) it appears in only works for ideal gases. Real gases deviate at high pressure and low temperature, where intermolecular forces and molecular volume matter. If you're working with steam at 200 atmospheres, PV = nRT is an approximation, not a truth.
Gotchas
- Celsius vs Kelvin. Thermodynamics formulas need absolute temperature. The ideal gas law (PV = nRT) takes T in Kelvin. Heat transfer (Q = mcΔT) takes ΔT — and because it's a difference, Celsius and Kelvin give the same answer (a 10°C change is a 10 K change). But if the formula has T as an absolute value, not a difference, you need Kelvin. Carnot efficiency (η = 1 − T_c/T_h) only works with absolute temperatures — plug in Celsius and you get efficiency above 100%, which violates thermodynamics.
- Scalar vs vector. Energy and power are scalars — they have magnitude only. Force, velocity, and momentum are vectors — they have magnitude and direction. Adding two forces in opposite directions gives a smaller net force, not a bigger one. The tool computes magnitudes; if your problem has directional components, resolve them into x and y components first, then apply the formula to each axis separately.
- The work angle. W = F·d·cos(θ) — the angle is between the force and the displacement, not between the force and the horizontal. If you're pulling a sled at 30° above horizontal and the sled moves horizontally, θ = 30°. If you're pushing a lawn mower forward and the handle is at 45°, θ = 45°. Getting this wrong gives you work that's too high or too low by a factor of cos(θ).
- Series vs parallel resistance. Resistors in series add directly (R_total = R₁ + R₂ + ...). Resistors in parallel add as reciprocals (1/R_total = 1/R₁ + 1/R₂ + ...). Two 100 Ω resistors in series give 200 Ω. In parallel they give 50 Ω. The tool lists both formulas, and mixing them up is the most common error in introductory circuits.
- Sign conventions in optics. The thin lens equation (1/f = 1/d_o + 1/d_i) uses sign conventions: object distance is positive for real objects, image distance is positive for real images and negative for virtual ones, focal length is positive for converging lenses and negative for diverging ones. The tool's solver doesn't handle signs — it does the arithmetic. You need to know which values are negative before entering them.
Summary
- The five formula families are kinematics (motion, no forces), dynamics (forces cause motion), energy (scalar conservation), momentum (vector conservation), and fields (inverse square laws). Knowing the family tells you which formula to reach for.
- Unit consistency is the #1 source of wrong answers. Every formula in the tool specifies the units each variable expects. Convert before you plug in.
- g is 9.81 m/s² on Earth, not everywhere. The ideal gas law only works for ideal gases. Know which constants are truly constant and which are situational.
- Use the Physics Formulas tool for the formula reference and solve-for-any-variable calculator, the Unit Converter for getting your inputs into the right units, and the Percentage Calculator / BMI Calculator for the other formula-based calculations.