Same scalar-versus-vector split you met with distance and displacement — now it’s about how fast you’re going. Speed tells you the rate you cover ground; velocity tells you that rate and the direction you’re heading. Two cars can share a speedometer reading of 20 m s−1 yet have completely different velocities if they’re pointing opposite ways.
📘 What you need to know
Speed = distance travelled per second — a scalar (magnitude only)
Velocity = rate of change of displacement — a vector (magnitude + direction)
Average speed = total distance ÷ time; average velocity = displacement ÷ time
Velocity can be negative — a sign shows the direction along an axis
Instantaneous speed/velocity is the value at a single moment
On a displacement–time graph, instantaneous velocity is the gradient of the tangent
Speed — the scalar
The speed of an object is the distance it travels every second. It’s a scalar — only a magnitude, no direction attached. The average speed over a journey is found from the total distance and the time taken.
Average speed
average speed = total distance ÷ time taken
The SI unit is metres per second (m s−1), though you’ll sometimes meet km h−1 or mph.
Velocity — the vector
Velocity is like speed but carries a direction. Formally it’s defined as the rate of change of displacement, which makes it a vector — magnitude and direction both. Because direction is baked in, velocity can take a negative value: a ball thrown up at 3 m s−1 comes back down at −3 m s−1 if up is positive, even though its speed is 3 m s−1 both ways.
Average velocityv̄ = s ÷ Δt
where s is total displacement (m) and Δt is the total time (s). If acceleration is constant and the initial velocity u and final velocity v are known, the average velocity is also just their midpoint, (u + v) ÷ 2.
distance
÷ time →
speed (scalar)
displacement
÷ time →
velocity (vector)
The two-cars example is worth holding onto: both read 20 m s−1 on the speedo, but one is 20 m s−1east and the other 20 m s−1west. Same speed, opposite velocities. The moment a question gives you a direction — or a sign — it’s talking about velocity, not speed. Don’t drop that sign; it’s usually doing real work later in the problem.
Instantaneous speed & velocity
The instantaneous speed or velocity is the value at a single instant in time, rather than averaged over a whole journey. On a displacement–time graph, a straight line means constant velocity, while a curved line means the velocity is changing — the object is accelerating.
To read the instantaneous velocity off a curved displacement–time graph, draw a tangent at the moment you care about, then find the gradient of that tangent.
Instantaneous velocity at a moment is the gradient of the tangent drawn to the displacement–time curve at that point.
Quick recap: straight displacement–time line → constant velocity (read the slope directly); curved line → changing velocity (draw a tangent, then take its gradient).
WE 1
A sprinter runs 200 m in a time of 21.34 s. Calculate her average speed.
Use average speed = distance ÷ time
her speed changes through the race, so we average it
average speed = 200 ÷ 21.34 = 9.372…average speed = 9.37 m s⁻¹Speed is a scalar — no direction needed here.
WE 2
On the displacement–time graph above, a tangent drawn at t = 9 s rises from s = 0 m to s = 10 m as time goes from 5 s to 13 s. Estimate the instantaneous velocity.
Instantaneous velocity = gradient of the tangent
gradient = rise ÷ run = Δs ÷ Δt
= (10 − 0) ÷ (13 − 5) = 10 ÷ 9velocity ≈ 1.1 m s⁻¹Read the rise and run off a large triangle for accuracy — a small triangle magnifies errors.
WE 3
A jogger runs 30 m east in 10 s, then straight back 30 m west in another 10 s. Find the average speed and the average velocity for the whole trip.
Average speed — uses total distance= (30 + 30) ÷ 20 = 3.0 m s⁻¹Average velocity — uses displacement
they end where they started, so displacement = 0
= 0 ÷ 20average velocity = 0 m s⁻¹Same journey, two answers — the classic scalar-vs-vector trap.
💡 Top tips
Draw a tangent that just touches the curve — the angle between curve and tangent should look equal on both sides of the point.
Use a large gradient triangle — the bigger the rise and run, the smaller your reading error.
Keep the sign of velocity — a negative value means motion in the negative direction, not an error.
If a question names a time (“velocity at t = 4 s”), it wants the instantaneous value — reach for a tangent.
⚠ Common mistakes
Using distance to find velocity — velocity comes from displacement
Dropping the direction or sign, turning a velocity into a bare speed
Reading the gradient off a tiny triangle, magnifying the error
Confusing the average value over a trip with the instantaneous value at one moment
Up next: Acceleration — the rate of change of velocity. Now that velocity is a vector, changing its size or its direction counts as accelerating, and a negative value tells its own story.
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