Measurement, Units, and Mathematical Tools
Establish the foundations of physics through SI units, dimensional analysis, significant figures, vector mathematics, and graphical interpretation.
Learning Objectives
- 1Convert between SI units using dimensional analysis and metric prefixes
- 2Apply significant figure rules to measurements and calculations
- 3Resolve vectors into components and add vectors using trigonometry
- 4Interpret position-time and velocity-time graphs to extract physical meaning
Before you learn how to measure anything, watch Veritasium dismantle what you THINK physics is. Most students believe physics is about memorizing formulas. This video reframes it as a way of thinking -- which is exactly the mindset you need for this course.
Watch on YouTubeWhy Measurement Matters
Physics is the science of how the universe works, and measurement is how we interrogate the universe. Every claim in physics must be testable, and every test requires measurement. Without a shared language of units, no experiment would be reproducible and no result would be meaningful.
The International System of Units (SI) gives us that shared language. Seven base quantities anchor all of physics:
| Base Quantity | SI Unit | Symbol |
|---|---|---|
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | K |
| Electric current | ampere | A |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Every other unit in physics is built from combinations of these seven. Speed is meters per second (m/s). Force is kilograms times meters per second squared (kg m/s^2), which we call a newton. Energy is kg m^2/s^2, which we call a joule.
Think About
The kilogram is the only SI base unit that contains a prefix (kilo-). A gram was originally defined as the mass of one cubic centimeter of water, but that turned out too small for practical standards. What might go wrong if we tried to define all base units from everyday objects rather than fundamental physical constants?
Dimensional Analysis
Dimensional analysis is the physicist's most reliable tool for checking work. Every valid physics equation must be dimensionally consistent -- the units on the left side must match the units on the right.
Example: Convert 95 km/h to m/s.
95 km/h x (1000 m / 1 km) x (1 h / 3600 s) = 26.4 m/s
Each conversion factor equals 1, so multiplying by it does not change the physical quantity -- only the units.
Example: Is the equation v = v_0 + at dimensionally consistent?
- Left side: v has dimensions of [length/time]
- Right side: v_0 has dimensions of [length/time], and at has dimensions of [length/time^2] x [time] = [length/time]
- Both terms on the right have the same dimensions as the left. The equation is dimensionally valid.
Dimensional analysis cannot tell you whether an equation is physically correct -- only whether it is dimensionally possible. The equation v = v_0 + 3at has correct dimensions but is wrong. Always check dimensions first, then check the physics.
Significant Figures
Every measurement has limited precision. Significant figures communicate how precisely a quantity is known.
Rules for counting significant figures:
- All nonzero digits are significant: 425 has 3 sig figs
- Zeros between nonzero digits are significant: 4005 has 4 sig figs
- Leading zeros are NOT significant: 0.0032 has 2 sig figs
- Trailing zeros after a decimal point ARE significant: 2.50 has 3 sig figs
Calculation rules:
- Multiplication/division: Round to the fewest sig figs among your inputs
- Addition/subtraction: Round to the fewest decimal places among your inputs
Example: A car travels 125.4 m in 3.2 s. What is its speed?
v = 125.4 / 3.2 = 39.1875 m/s
The input with the fewest sig figs is 3.2 (two sig figs), so the answer rounds to 39 m/s.
Think About
When NASA calculates a trajectory to Mars, they use 15 or more significant figures. A small rounding error at launch becomes a massive miss millions of kilometers later. But when you measure the length of a table with a meter stick, 3 sig figs is plenty. How do you decide how much precision a problem actually needs?
Scalars and Vectors
Physics deals with two kinds of quantities:
- Scalars have magnitude only: temperature (25 degrees C), mass (4.5 kg), speed (12 m/s)
- Vectors have both magnitude and direction: displacement (30 m east), velocity (12 m/s north), force (50 N at 30 degrees above horizontal)
Vectors are represented by arrows where the length represents magnitude and the direction of the arrow represents the direction of the quantity.
Adding Vectors: Component Method
To add vectors that are not along the same line, resolve each into perpendicular components.
For a vector A with magnitude A at angle theta above the positive x-axis:
- A_x = A cos(theta)
- A_y = A sin(theta)
Example: A hiker walks 5.0 km at 37 degrees north of east, then 3.0 km due north. Find the resultant displacement.
Step 1 -- Resolve into components:
- Walk 1: A_x = 5.0 cos(37) = 4.0 km, A_y = 5.0 sin(37) = 3.0 km
- Walk 2: B_x = 0 km, B_y = 3.0 km
Step 2 -- Add components:
- R_x = 4.0 + 0 = 4.0 km
- R_y = 3.0 + 3.0 = 6.0 km
Step 3 -- Find magnitude and direction:
- R = sqrt(4.0^2 + 6.0^2) = sqrt(52) = 7.2 km
- theta = arctan(6.0/4.0) = 56 degrees north of east
The component method for vector addition relies on right triangle trigonometry -- the same sine, cosine, and tangent ratios you studied in geometry. Every vector problem is ultimately a triangle problem.
Graphs in Physics
Graphs are not just pictures -- they encode physical relationships. Learning to read a graph is as important as reading an equation.
Position-Time Graphs
- Slope = velocity (rise/run = change in position / change in time)
- Straight line = constant velocity
- Curved line = changing velocity (acceleration)
- Horizontal line = at rest
Velocity-Time Graphs
- Slope = acceleration
- Area under the curve = displacement
- Horizontal line = constant velocity (zero acceleration)
- Straight line with nonzero slope = constant acceleration
Think About
A velocity-time graph shows a straight line that starts at 10 m/s and drops to -5 m/s over 3 seconds. The object's speed decreases, passes through zero, and then the object moves backward. At what time does the object momentarily stop? What does the area under the line represent physically?
❓Concept Check
A position-time graph shows a parabola opening upward. What does this tell you about the object's velocity and acceleration?
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Concept Check
A position-time graph shows a parabola opening upward. What does this tell you about the object's velocity and acceleration?
A parabola on a position-time graph means the position is changing as a quadratic function of time, which indicates constant acceleration. The velocity (slope of the tangent line) is increasing linearly with time, and the acceleration is constant and positive.
Looking Ahead
With measurement, vectors, and graphs in your toolkit, you are ready to tackle kinematics -- the mathematical description of motion. In the next unit, you will derive the equations that describe how objects move in one dimension, from cars on highways to balls in free fall.
