Problem — Rome-Florence average and instantaneous

Problem

Average velocity and instantaneous velocity. A car goes from Rome to Florence in $3$ hours, covering $250\;\text{km}$. Explain in five lines the difference between average velocity and instantaneous velocity, and what exactly a speed camera measures.

Solution

Average velocity is the ratio between the distance covered and the time taken, without caring about what happens in between:

$\bar v = \frac{\Delta x}{\Delta t} = \frac{250\;\text{km}}{3\;\text{h}} \approx 83\;\text{km/h}$

This does not mean the car travelled at $83\;\text{km/h}$ the whole time: on the motorway it will have touched $130$, at the toll booths it will have stopped. The average flattens all these variations into a single number.

Instantaneous velocity, instead, is the value at a single instant, defined as the derivative of position, $v = dx/dt$: it is what the speedometer shows moment by moment.

A speed camera measures precisely the instantaneous velocity (or nearly so: it is estimated over a very short stretch) at the point where it is installed. This is why you can respect the average speed for the whole journey and still get fined: what counts is the velocity at the instant and point of detection, not the average of the entire route.

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Topics: Cinematica Concepts: Velocità media