Skills Beta

Math you need: the one-page sheet

Units, Prefixes and Significant Figures

Work in SI units (m, kg, s). Convert by multiplying by fractions equal to 1 so the unwanted unit cancels, use each factor twice for an area, and keep as many significant figures as the least precise data, rounding once at the end.

Physics 1 works in SI units: the meter (m), kilogram (kg) and second (s), and units built from them such as m/s.
A prefix is a power of ten: kilo = 10³, centi = 10⁻², milli = 10⁻³, micro = 10⁻⁶. So 1 km = 1000 m, 1 cm = 0.01 m, 1 g = 0.001 kg.
To convert, multiply by a fraction equal to 1 with the unit you want on top and the unit you are removing on the bottom. Check that the units cancel.
Areas need the factor twice: 1 m² = 100 cm × 100 cm = 10 000 cm².
Significant figures: multiplying or dividing, keep the fewest significant figures of the data; adding or subtracting, the fewest decimal places. Round once, at the end.

nobody can tell how big the quantity is, so a value is always a number times a unit the units of the answer come out in SI too, with no hidden factors of 100 or 1000 the value does not change, but the unwanted unit cancels like a number a calculated result keeps the precision of the least precise data, rounded once at the end

SI unit: One of the agreed units of the International System. Physics 1 uses the base units meter (m) for length, kilogram (kg) for mass and second (s) for time, and units built from them, such as m/s.
metric prefix: A word in front of a unit that multiplies it by a power of ten: kilo (k) = 10³, centi (c) = 10⁻², milli (m) = 10⁻³, micro (μ) = 10⁻⁶.
unit conversion: Changing a value to another unit by multiplying by a fraction equal to 1, such as (1 m / 100 cm), so the old unit cancels.
scientific notation: Writing a number as a value from 1 to 10 times a power of ten, such as 2.5 × 10⁻³ s for 0.0025 s.
significant figures: The digits of a measured value that are known, plus one estimated digit. A calculated result keeps as many as the least precise data allow.

Rearranging Equations with Symbols

Symbols stand for quantities with units, subscripts are labels and Δ means final minus initial. Rearrange by undoing operations in reverse order on both sides, solve in symbols first, then substitute numbers with units and check the units of the result.

A symbol is a quantity with a unit: x position (m), v velocity (m/s), t time (s). Subscripts are labels: x₀ is the starting position.
Δ means "change in" and is always final minus initial: Δx = x − x₀. A negative Δ means the quantity went down.
To rearrange, undo operations in reverse order, doing the same to both sides: from x = x₀ + vt, subtract x₀, then divide by v.
Solve in symbols first, then substitute numbers with units once. Check that the units of the result make sense.

you can check any rearranged equation by checking its units you can isolate any symbol by undoing operations in reverse order the result shows how the answer depends on each quantity, and you substitute only once a change can be negative, and its sign carries the direction

variable (symbol): A letter that stands for a physical quantity, such as x for position, v for velocity or t for time. It stands for a number with a unit.
delta notation: The Greek letter Δ (delta) in front of a symbol means "change in": final value minus initial value. Δx = x − x₀.
subscript notation: A small label written below a symbol to say which value is meant: x₀ is the starting position, v₁ and v₂ two different velocities.
rearranging an equation: Solving an equation for one symbol by doing the same operation to both sides, undoing the operations in reverse order.
symbolic expression: An answer written with symbols for given quantities instead of numbers, such as t = (x − x₀)/v. The exam asks for these in derivations.
substitution: Putting numbers, with their units, in place of the symbols of an expression, after it has been solved for the unknown.

Proportional Reasoning and Functional Dependence

Functional dependence says how one quantity responds when another changes. Directly proportional: the same factor. Inversely proportional: divide by the factor. Square: the factor squared. For several changes, multiply their factors.

To find a functional dependence, solve for the quantity asked, hold everything else fixed, and look at the power of the quantity that changes.
Directly proportional (y ∝ x): factor in, same factor out. Inversely proportional (y ∝ 1/x): factor in, divide by it.
Square (y ∝ x²): factor in, factor squared out. Triple x → 9y. A square root (y ∝ √x): quadruple x → 2y.
Several changes at once: write a factor for each and multiply them. In y = kx²/z, doubling x and z gives 2²/2 = 2.

the power of each quantity in the equation sets how the other responds multiplying x by a factor multiplies y by the same factor multiplying x by a factor divides y by that factor multiplying x by a factor multiplies y by the factor squared

directly proportional: y is directly proportional to x (y ∝ x) when y = kx for a constant k: multiplying x by any factor multiplies y by the same factor.
inversely proportional: y is inversely proportional to x (y ∝ 1/x) when y = k/x: multiplying x by a factor divides y by that factor, so x × y stays constant.
functional dependence: How one quantity changes when another changes, found from the equation that links them. Written as a factor: "if v doubles, d becomes 4 times as large".
square relationship: A dependence on the square of a quantity, y ∝ x²: multiplying x by a factor multiplies y by that factor squared. An inverse square, y ∝ 1/x², divides y by the factor squared.

Slopes and Areas of Graphs

Label every axis with a quantity and unit. The slope of a line is Δy/Δx between two points on it, with units (y unit)/(x unit); the area under a graph is height × width, with units (y unit) × (x unit). Fit a line to data, and linearize a curved relationship before fitting.

Every axis gets a label and a unit: "Time (s)". Without both, the graph cannot be read.
Slope = rise/run = Δy/Δx, from two points on the line (not data points). Its unit is (y unit)/(x unit), and it is a rate.
Area under a graph = height × width for a rectangle, ½ × base × height for a triangle. Its unit is (y unit) × (x unit).
A best-fit line passes between scattered points, never dot to dot. If the relationship is curved, linearize: for d = ½at², plot d against t², and the slope is ½a.

its slope and its area each have a unit, built from those two units you get the slope, the rate at which y changes per unit of x you get an area whose unit is (y unit) × (x unit), often a total such as a distance you draw a best-fit line, after choosing what to plot so the graph is straight

axis label: The labels on a graph's axes: the quantity and its unit, such as "Time (s)". The independent variable (the one you choose or control) usually goes on the horizontal axis.
slope: The steepness of a line: rise over run, Δy/Δx between two points on the line. Its unit is (y unit)/(x unit), and it tells you the rate at which y changes with x.
linear relationship: A relationship whose graph is a straight line, y = mx + b, with slope m and vertical intercept b.
area under a graph: The area between a graph and the horizontal axis over an interval. Its unit is (y unit) × (x unit); area below the axis counts as negative.
best-fit line: A straight line drawn through scattered data with about as many points above it as below. It is never drawn dot to dot; its slope uses all the data.
linearization: Choosing what to plot so that the graph is a straight line, for example d against t² when d = ½at², so the slope gives the quantity you want.

Right-Triangle Trigonometry and Vector Components

A vector at an angle is the hypotenuse of a right triangle. The component adjacent to the given angle is A cos θ and the opposite one is A sin θ; the magnitude is the square root of the sum of the squared components. Add vectors by components, with the calculator in degree mode.

SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
Components: the side adjacent to the given angle is A cos θ; the side opposite it is A sin θ. Check which axis the angle is measured from.
Back to the vector: magnitude = √(Aₓ² + A_y²); angle = tan⁻¹(A_y/Aₓ).
Set your calculator to degree mode. The sheet lists 37° and 53° (the 3-4-5 triangle): sin 37° ≈ 0.60, cos 37° ≈ 0.80.
Add vectors tip to tail, or by adding x-components and y-components separately.

it forms the hypotenuse of a right triangle whose legs lie along the axes the leg adjacent to the angle is A cos θ and the leg opposite it is A sin θ the magnitude is the square root of (x-component² + y-component²), and tan θ = y-component ÷ x-component you add their x-components and their y-components separately

right triangle: A triangle with one 90° angle. Its longest side, opposite the right angle, is the hypotenuse; the Pythagorean theorem says hypotenuse² = leg² + leg².
sine: For an angle θ in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH CAH TOA).
degree mode: The calculator setting that reads angles in degrees. Physics 1 angles are in degrees; in any other mode, sin and cos of a degree value come out wrong.
vector component: The parts of a vector along perpendicular axes: for magnitude A at angle θ from the x-axis, the x-component is A cos θ and the y-component is A sin θ. The vector is the sum of its components.
tip-to-tail addition: Adding vectors by drawing each one starting at the tip of the one before; the resultant runs from the first tail to the last tip. In practice, add the x-components and the y-components separately.