Speed and Velocity

The question

You run one lap of a 400 m track in 80 seconds, finishing right where you started. Your friend says you averaged 5 m/s. Your teacher says your average velocity was 0 m/s. Both are right — how?

Speed counts every metre you cover; velocity only cares where you finished versus where you started. End where you began, and your velocity averages to zero — the whole point of distance vs displacement.

The key idea

Speed is how much distance an object covers per unit time — a scalar, a single positive number. Velocity is how much displacement (the straight-line change in position, with direction) occurs per unit time — a vector, with both a magnitude and a direction.

$v=d\mathrm{/}t$ separates 'how fast' from 'how long', and rearranges to $d=v\cdot t$ and $t=d\mathrm{/}v$. The key split: average speed = total distance ÷ time (always positive), while average velocity = total displacement ÷ time (can be zero or negative). A round trip always gives average velocity 0 — exactly what you saw when the Displacement card returned to zero. **Limiting case:** any closed path — a lap, a round trip — has zero displacement, so average velocity is exactly zero however fast you ran; speed has no such collapse. **Connect it:** velocity is the slope of the position–time graph, and its own rate of change is the next lesson: acceleration. The chain $x\to v\to a$ is one idea differentiated twice.

The formula

• ·v = speed in metres per second (m/s)
• ·d = distance in metres (m)
• ·t = time in seconds (s). For velocity
• ·use displacement: v̄ = Δx / Δt
• ·where Δx is the change in position (which can be negative).

What to remember

• ·Speed is a scalar (distance ÷ time); velocity is a vector (displacement ÷ time).
• ·Average velocity depends only on start and end points, not the path taken.
• ·A complete round trip has zero displacement, so zero average velocity — however fast you went.