Average speed and instantaneous speed

Problem

Explain, with a concrete example, the difference between average speed and instantaneous speed. Then explain why a car stopped at a traffic light has zero acceleration but can still be subject to non-zero forces.

Solution

Average speed and instantaneous speed. The average speed over a time interval is the ratio between the total displacement and the duration of the interval:

$\bar{v} = \frac{\Delta s}{\Delta t}$

It describes the overall behaviour of the motion, without saying anything about what happens instant by instant. Instantaneous speed, on the other hand, is the value that the average speed tends to as the time interval becomes vanishingly small:

$v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}$

and expresses how fast the motion is at a precise instant.

Concrete example. A car covers $120\;\mathrm{km}$ in $2\;\mathrm{h}$: its average speed is $60\;\mathrm{km/h}$. This does not mean the car always travelled at $60\;\mathrm{km/h}$: on motorway stretches the speedometer (which shows instantaneous speed) may have read $130\;\mathrm{km/h}$, while stuck in traffic or at toll booths the same speedometer read $0\;\mathrm{km/h}$. The average summarises the whole journey; the instantaneous value captures a single moment.

Car stopped at a traffic light. A stationary car has constant speed (equal to zero) and therefore zero acceleration, because acceleration measures the change in speed over time. Zero acceleration does not, however, imply the absence of forces: it merely means that the resultant of the forces is zero. The car is in fact acted on by the weight force downwards and the normal reaction of the road surface upwards, equal and opposite; their vector sum is zero. It is balanced forces that produce zero acceleration, not the absence of forces.

$\ev{\bar{v} = \Delta s/\Delta t; \quad v = \lim_{\Delta t\to 0}\Delta s/\Delta t; \quad \text{forze bilanciate} \Rightarrow a = 0}$

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Topics: Cinematica Concepts: Velocità media