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Calculus I/Physics/Kinematics

Position, Velocity, Acceleration

Velocity is how fast position changes. Acceleration is how fast velocity changes. Each one is the derivative of the one before it.

Try it yourself

Position s(t) =

Time window4.0 s

velocity

v = s'

acceleration

a = v'

turns around

How it works

Derivatives as rates of change

In one sentence

If $s(t)$ is position, velocity is its derivative $v = s'$ and acceleration is $a = v' = s''$ .

General formula

{\htmlData{sym=1}{v}}({\htmlData{sym=3}{t}}) = {\htmlData{sym=0}{s}}'({\htmlData{sym=3}{t}}), \qquad {\htmlData{sym=2}{a}}({\htmlData{sym=3}{t}}) = {\htmlData{sym=1}{v}}'({\htmlData{sym=3}{t}}) = {\htmlData{sym=0}{s}}''({\htmlData{sym=3}{t}})

A derivative is the slope of a graph: how much position changes per second right now.

What the symbols mean

$s(t)$
Position
Where the car is along the track.
$v$
Velocity
How fast the position changes, with a sign for direction.
$a$
Acceleration
How fast the velocity changes. Pressing the gas or the brake.
$t$
Time
Seconds since the car started.

The slope of the tangent line to $s(t)$ at an instant is the velocity at that instant.
When $v(t) = 0$ the object is momentarily stopped, often turning around.
Acceleration pointing against velocity means slowing down.
Going backwards, the area under $v(t)$ is the displacement.

Learn the math

Tangent Lines and Rates of ChangeCalc I · Lesson 4, Limits The Definition of the DerivativeCalc I · Lesson 14, Derivatives Interpreting the DerivativeCalc I · Lesson 15, Derivatives Higher Order DerivativesCalc I · Lesson 25, Derivatives Indefinite IntegralsCalc I · Lesson 40, Integrals