Units, Measurement, and Vectors

Imagine you are part of a mission team landing a probe on Mars. The retro-rockets fire for a number that one engineer wrote down as 8.4 — but 8.4 of what? Seconds? Tenths of a second? In 1999 NASA lost the 327 million dollar Mars Climate Orbiter because one team worked in pound-seconds while another assumed newton-seconds. Physics begins with a discipline that sounds mundane and turns out to be life-or-death: every quantity carries a unit, and every direction matters.

Units and the SI system

Physics describes the world with measured quantities, and a measurement is meaningless without a unit. Mechanics needs only three base units, all from the International System (SI): the meter (m) for length, the kilogram (kg) for mass, and the second (s) for time. Everything else in this course is built from these. Speed is meters per second (m/s), acceleration is meters per second squared (m/s²), and force will turn out to be kg·m/s², a combination so common it gets its own name, the newton.

Units are also a powerful error detector. The technique called dimensional analysis checks whether an equation can possibly be right: both sides must carry identical units. If you derive a formula for a distance and the right-hand side works out to m/s, you have made a mistake — no algebra required to know it. Get in the habit of carrying units through every line of every calculation.

Scalars and vectors

Some quantities are fully described by a single number with a unit: temperature, mass, time, speed. These are scalars. Others are incomplete without a direction: displacement, velocity, acceleration, force. These are vectors, drawn as arrows whose length represents magnitude. Walking 5 km matters very differently depending on whether it is 5 km toward the trailhead or 5 km off a cliff edge — the direction is part of the physics.

The key skill is breaking a vector into perpendicular components. A vector A→ at angle θ above the x-axis has components Aₓ = A cos θ and A_y = A sin θ. To add vectors, add their components separately, then rebuild the total: the magnitude is A = √(Aₓ² + A_y²) and the direction is θ = arctan(A_y / Aₓ). Components turn geometry into arithmetic, and they are the reason two-dimensional problems split into two independent one-dimensional problems — an idea we will exploit constantly.

Worked example: a hiker's displacement

A hiker walks 3.0 km due east, then 4.0 km due north. What is her total displacement — magnitude and direction?

Set up axes with x pointing east and y pointing north. The two legs are already components: Δx = 3.0 km and Δy = 4.0 km.

magnitude:  d = √(Δx² + Δy²) = √(3.0² + 4.0²) km
              = √(9.0 + 16.0) km = √25.0 km = 5.0 km

direction:  θ = arctan(Δy / Δx) = arctan(4.0 / 3.0)
              = arctan(1.333) = 53.1° north of east

Notice what displacement means: although she walked 3.0 + 4.0 = 7.0 km of trail (a scalar distance), her displacement — the straight-line vector from start to finish — is only 5.0 km at 53.1° north of east. Distance and displacement are different quantities, and physics almost always cares about the vector.

Why this matters

Everything in this course — projectiles, forces, momentum, torque — is a vector story told in components, with units as the grammar. Engineers checking a bridge load, pilots correcting for crosswind, and game developers writing a physics engine all begin exactly here: choose axes, resolve into components, keep the units honest. Master this lesson and every later one gets easier.

Curriculum aligned with OpenStax's University Physics Volume 1; all lesson text is original to Syllabus.