Kinematics: Position, Velocity, and Acceleration

Kinematics describes motion without yet asking what causes it.

Motion dots aligned with position, velocity, and acceleration graphs showing slope and area relationships. — Local explanatory diagram

Three quantities form the core:

position $x$ : where an object is
velocity $v$ : how fast position is changing, including direction
acceleration $a$ : how fast velocity is changing

In one dimension:

v = Δx/Δt

for an average over a finite interval, while the instantaneous version is

v = dx/dt.

Likewise,

a = Δv/Δt

on average, and

a = dv/dt

instantaneously.

The graph hierarchy

Kinematics becomes much clearer when graphs are read geometrically.

On a position-versus-time graph:

slope = velocity

On a velocity-versus-time graph:

slope = acceleration
signed area = displacement

On an acceleration-versus-time graph:

signed area = change in velocity

That relationship is more general than any memorized constant-acceleration formula.

Constant acceleration

If acceleration is constant:

v = v₀ + at

x = x₀ + v₀t + 1/2at²

v² = v₀² + 2a(x-x₀)

where:

$x₀$ = initial position
$v₀$ = initial velocity
$a$ = constant acceleration
$t$ = elapsed time

These equations are convenient, but their constant-acceleration assumption matters.

A car accelerating hard from rest does not generally maintain the same acceleration all the way to highway speed. Air drag makes a falling object's acceleration depart from $g$ as speed increases. The equations above stop being exact when $a$ varies significantly.

Velocity and acceleration need not point the same way

A ball thrown upward has upward velocity while gravity gives it downward acceleration.

At the very top:

v = 0

for an instant, but

a \approx -9.8 m/s²

near Earth's surface.

Zero velocity therefore does not imply zero acceleration.

Two dimensions: treat vectors as vectors

Projectile motion can be decomposed into components.

Ignoring air resistance near Earth's surface:

ax = 0

ay = -g.

Horizontal and vertical motions share the same time coordinate but otherwise follow their own component equations.

This is why a horizontally fired projectile and a dropped object released from the same height hit the ground at the same time, assuming the ground is level and air resistance is negligible.

Physics C bridge

Calculus removes the need for constant acceleration.

Given acceleration as a function of time:

v(t)=v(t₀)+\intt_₀ta(t') dt'

and then:

x(t)=x(t₀)+\intt_₀tv(t') dt'.

Differentiation goes the other way:

x(t) \to v(t) \to a(t).

The familiar kinematic equations are what these integrals produce when $a$ is constant.

Takeaway: Position, velocity and acceleration are not three formulas. They are a derivative/integral hierarchy describing how motion changes.

Quantitative anchor

At $9.8 m/s²$ , an object in ideal free fall changes its downward velocity by about 9.8 m/s every second — roughly 22 mph of speed change per second.

Misconceptions

Negative velocity does not mean slowing down.
Negative acceleration does not automatically mean slowing down.
A flat position graph means zero velocity, not zero position.
A turning point can have $v=0$ while $a\neq0$ .

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sources

College Board — AP Physics 1 Course and Exam Description College Board — AP Physics 1 equation sheet OpenStax University Physics Volume 1