Position, velocity, acceleration.

The three functions that describe motion along a line, and how each is the time-derivative of the one before it.

The vocabulary

Along a straight line, every object's motion can be described by three time-dependent functions:

x(t) — position (meters) v(t) = dx/dt — velocity (meters per second) a(t) = dv/dt = d²x/dt² — acceleration (m/s²)

Each is the time-derivative of the previous. This is no accident — calculus was invented to describe motion, and these are the primitive concepts it was built on.

Constant acceleration — the SUVAT equations

When acceleration is constant, the integrals collapse to four equations that cover nearly every introductory mechanics problem. They relate five quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).

v = u + at s = ut + ½ a t² v² = u² + 2as s = ½ (u + v) t

Pick the equation that contains the three quantities you know and the one you want — the missing fourth is what you solve for.

Worked example

A car starts from rest (u = 0) and accelerates uniformly at 2 m/s² for 5 seconds. How far does it travel?

s = ut + ½ a t² s = (0)(5) + ½ (2)(5)² s = 0 + ½ · 2 · 25 = 25 m

The car travels 25 m in those 5 seconds. The initial velocity was zero, so the first term vanishes — only the acceleration term contributes.

Flashcard · Unit 2

F = m · a

A net force on a mass produces acceleration in the same direction.

Check your understanding

A ball is dropped from rest. After 3 seconds (ignoring air resistance), what is its velocity? Take g = 9.8 m/s².