Velocity, Speed & Acceleration

Your car's speedometer says 60. Your GPS says you're heading north. One is a scalar, one is a vector, and confusing them is the single most common slip in introductory physics.

Speed, velocity and acceleration are the three measurements that describe how things move. They sound almost interchangeable in everyday conversation — but in physics they mean precisely different things. Speed is how fast you're going. Velocity is how fast and in which direction. Acceleration is how quickly your velocity changes. Get the distinctions straight and a huge chunk of mechanics suddenly clicks.

Speed: The Scalar Quantity

Speed measures how much distance an object covers in a given time, with no regard for direction. It's a scalar quantity, meaning it has size but no orientation. The formula is simple:

If you walk 100 metres in 50 seconds, your average speed is 2 m/s. The speedometer in a car reads instantaneous speed — how fast you're going at that exact moment, not an average over the whole journey.

Average speed vs instantaneous speed

• Average speed = total distance ÷ total time. On a 300 km road trip that takes 5 hours, average speed is 60 km/h — even if you hit 120 km/h on the motorway and 0 km/h at traffic lights.
• Instantaneous speed = how fast you're going at one specific instant, what a speedometer displays.

Velocity: Speed With Direction

Velocity is speed in a specified direction. It's a vector quantity, which means both a number and a direction are essential to describe it fully. The fundamental definition uses displacement, not distance:

Distance vs displacement — the key to understanding velocity

Distance is the total ground covered, a scalar. Displacement is how far you are from where you started, in a specific direction — a vector. Walk 3 metres east, then 3 metres west: you've covered a distance of 6 metres, but your displacement is zero. Your speed was positive throughout, but your average velocity was zero, because you ended where you began.

When speed is constant but velocity isn't

A car driving around a roundabout at a steady 30 km/h has constant speed but changing velocity, because its direction is continuously changing. This is a crucial distinction: velocity depends on direction, so any change in direction means velocity changes — which means the object is accelerating.

Acceleration: The Rate of Change of Velocity

Acceleration is how quickly velocity changes over time. Like velocity, it's a vector. The formula is:

Acceleration is measured in metres per second squared (m/s²). A value of 5 m/s² means the velocity increases by 5 m/s every second.

Three ways an object can accelerate

1. Speeding up — velocity and acceleration point in the same direction. A car pulling away from a stop sign.
2. Slowing down (deceleration) — velocity and acceleration point in opposite directions. A car braking at a red light.
3. Changing direction — even at constant speed. A ball swung in a circle on a string is accelerating because its direction keeps changing.

Worked example: from 0 to 100 km/h

A car accelerates from rest to 100 km/h (27.8 m/s) in 5 seconds. What's its acceleration?

1. Convert units: 100 km/h ÷ 3.6 = 27.8 m/s.
2. Use the formula: a = (v − u) / t
3. Substitute: a = (27.8 − 0) ÷ 5
4. Solve: a = 5.56 m/s²

Speed vs Velocity vs Acceleration: Side-by-Side

Here's the clearest way to see how these three quantities relate — and where each one fits.

Quick check: which one changes?

Reading Motion Graphs

Motion graphs encode these relationships visually. The slope of a velocity-time graph gives acceleration; the area under it gives displacement. The slope of a distance-time graph gives speed.

• Flat horizontal line on a velocity-time graph → zero acceleration (constant velocity).
• Straight sloping line → uniform (constant) acceleration.
• Line crossing below the time axis → motion in the opposite direction (negative velocity).
• Curved line → changing acceleration (non-uniform acceleration).

Real-World Examples You Already Know

These distinctions aren't academic — they show up in everyday situations all the time.

• A plane circling before landing: constant speed (the airspeed indicator stays steady), but velocity is constantly changing because the direction changes. The plane is accelerating, and passengers feel that acceleration as a gentle push toward the outside of the turn.
• An elevator starting and stopping: when it begins moving upward, you feel heavier — that's upward acceleration. When it slows at your floor, you feel lighter — downward acceleration. At constant speed in between, you feel normal weight.
• A ball thrown straight up: at the highest point, its velocity is instantaneously zero, but its acceleration is not. Gravity is still pulling it down at 9.8 m/s², which is why it immediately starts falling.
• A car's speedometer: shows speed, not velocity. Two cars side by side at 50 km/h have the same speed but different velocities if they're heading in different directions.

Three Mistakes Almost Everyone Makes

1. Using speed and velocity interchangeably. In everyday chat, it's fine. In a physics problem, it'll cost you marks. Velocity needs a direction — always.
2. Thinking acceleration means "going faster." It means velocity is changing. Braking is acceleration. Turning is acceleration. Only steady straight-line motion has none.
3. Assuming constant speed means zero acceleration. A car going around a bend at a steady 40 km/h is accelerating, because direction is part of velocity. The speedometer might not move, but the accelerometer would show a reading.