Position, Velocity, and Acceleration

By the end of this lesson, you will be able to define and distinguish position, distance, displacement, speed, velocity, and acceleration, and analyze motion using position-time and velocity-time graphs.

The Language of Motion (Kinematics)

Position ($x$): location of an object relative to a reference point. Vector quantity (in 1D, sign indicates direction).

Distance: total path length traveled. Scalar.

Displacement ($\Delta x = x_f - x_i$): change in position. Vector — can be negative.

Example: A person walks 6 m east, then 3 m west. Distance = 9 m; Displacement = 3 m east.

Speed and Velocity

Average speed: total distance / total time. Scalar.

Instantaneous speed: magnitude of velocity at an instant.

Average velocity: $\bar{v} = \Delta x / \Delta t$. Vector.

Instantaneous velocity: $v = dx/dt$ (the derivative of position). The slope of the $x$-$t$ graph.

Acceleration

Average acceleration: $\bar{a} = \Delta v / \Delta t$. Vector.

Instantaneous acceleration: $a = dv/dt$. The slope of the $v$-$t$ graph.

Positive acceleration: velocity increasing in positive direction (or decreasing in negative direction).

Negative acceleration (deceleration): velocity decreasing in positive direction.

Note: acceleration and velocity can be in opposite directions → slowing down.

Position-Time ($x$-$t$) Graphs

Slope of $x$-$t$ graph = instantaneous velocity.

• Positive slope: moving in positive direction
• Negative slope: moving in negative direction
• Zero slope: at rest
• Curved line: changing velocity (acceleration)

Velocity-Time ($v$-$t$) Graphs

Slope of $v$-$t$ graph = acceleration.

Area under $v$-$t$ graph = displacement (positive area = positive displacement).

• Constant positive slope: constant positive acceleration
• Decreasing positive velocity: negative acceleration (decelerating)
• $v$-$t$ crosses zero: object reversed direction

Example 1 — Reading a $v$-$t$ Graph

A car's velocity vs. time: starts at 0, increases to 20 m/s over 5 s, then stays at 20 m/s for 3 s, then decreases to 0 in 2 s.

$a = \Delta v/\Delta t = 20/5 = 4$ m/s²