Q.Explain, with reasoning, why the average speed of a body over a given time interval can never be less than the magnitude of its average velocity over the same interval.
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Imagine you're walking home from school. You take a shortcut through a park, then stop to buy a snack, then realise you forgot something and run back a bit, then finally walk home. By the time you reach your front door, you've walked a total of 2 km — but your house is only 500 metres from school in a straight line.
That difference — between the total ground you covered and how far you actually ended up from where you started — is the entire point of speed vs velocity.
The Precise Definitions
Average speed is a measure of how fast something is moving overall. It cares only about the total distance travelled, not the direction.
Average speed=Total time takenTotal distance travelled
Average velocity is a measure of how fast and in what direction something is moving overall. It cares about the net displacement — the straight-line distance from start to finish, with a direction.
Average velocity=Total time takenDisplacement
Note
Displacement is the straight-line distance from the starting point to the ending point, with a direction. Distance is the total length of the actual path travelled, with no direction.
The Key Difference in One Sentence
Speed is a scalar (just a number, like 5 km/h). Velocity is a vector (a number and a direction, like 5 km/h north).
That one word — direction — changes everything.
A Concrete Example
You walk 3 km east, then 4 km north. The whole trip takes 1 hour.
Total distance travelled = 3 + 4 = 7 km
Displacement = straight line from start to finish = 32+42=5 km, northeast
Now compute:
Average speed=1 h7 km=7 km/h
Average velocity=1 h5 km, northeast=5 km/h, northeast
Watch out
A common mistake: students think average velocity is just "speed with direction". It's not. It's displacement divided by time, not distance divided by time. If you walk in a circle and return to your starting point, your displacement is zero — so your average velocity is zero, even though your average speed is positive.
When Are They Equal?
Only when the motion is in a straight line without changing direction. If you walk 2 km east in a straight line, then distance = displacement, so average speed = magnitude of average velocity.
But the moment you turn, or stop, or go backwards — they diverge.
Why This Matters for Exams
In Indian board exams (CBSE, ICSE, state boards), you will be asked to:
Distinguish between speed and velocity (scalar vs vector)
Calculate average speed and average velocity from given data …
[!TLDR] Path length is always ≥ the magnitude of displacement over the same interval, and dividing both by the same time gives average speed ≥ |average velocity|. [!ANSWER] Average speed ≥ |average velocity| always, because p …
The geometric fact. For any interval of straight-line motion, the path length (the total length of the actual route travelled) can never be less than the magnitude of the displacement (the net, straight-line change in position) over that same interval:
path length≥∣Δx∣
This is because displacement measures only the net effect of the motion, while path length additionally accounts for any distance covered in reversing or retracing part of the route — any such reversed distance adds to path length but partially or wholly cancels out of the net displacement.
Dividing by the same time interval. Average speed and (the magnitude of) average velocity are defined by dividing path length and ∣Δx∣, respectively, by the SAME time interval Δt (Section 2.3):
average speed=Δtpath length,∣vˉ∣=Δt∣Δx∣
Since the numerator of average speed is always at least as large as the numerator of ∣vˉ∣, and both are divided by the same positive Δt, it follows immediately that
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2025Set ANNUAL1 markMCQ
Q.A cyclist moving on a circular track of radius 40 m completes half a revolution in 40 s. Its average velocity is
(a) zero
(b) 2 m s^-1
(c) 4 pi m s^-1
(d) 8 pi m s^-1
›Reveal solutionSolution
Average velocity uses DISPLACEMENT, not distance travelled; half a revolution displaces the cyclist by one diameter (2r), giving 2 m/s.
Radius r = 40 m, so diameter = 2r = 80 m.
In half a revolution, the cyclist moves from one end of a diameter to the exact opposite end of the circle. The straight-line displacement between these two points equals the diameter, 80 m (NOT the arc length, which is used for average SPEED, not average velocity).
Q.In 1.0 second, a particle goes from point A to point B moving in a semi-circle of radius 1.0m as shown in fig. The magnitude of average velocity is
(a) 3.14 m/s
(b) 2.0 m/s
(c) 1.0 m/s
(d) Zero
›Reveal solutionSolution
Average velocity uses displacement (straight-line distance A to B = diameter), giving 2.0 m/s, not the arc length (which would give average SPEED = 3.14 m/s).
The particle moves along a semicircular arc of radius r = 1.0 m from A to B in time t = 1.0 s.
Displacement (straight line from A to B) = diameter = 2r = 2 x 1.0 = 2.0 m.
Average velocity = displacement / time = 2.0 m / 1.0 s = 2.0 m/s.
