Counting trucks

I have a confession to make: I am curious, but I am also a bit lazy, so the other day I wanted to know how many trucks per hour were driving along the highway near where I live.

I could walk there of course, but that is quite a walk, but since I use that same highway when I drive to work, I was wondering if I could count those trucks while driving my car and still get the result I am looking for.

If you were standing still near the road, it would be easy: just count every truck as it passes for five minutes, multiply by twelve, and voilá, you get the number of trucks per hour.

But what if you were driving in the opposite direction at the same speed as the trucks? In Europe trucks have mandatory speed limiters restricting them to eighty kilometers per hour, so naturally they all drive at this maximum speed if they can.

If you drive in the opposite direction at the same speed you would count double the number of trucks, because in the time it takes for a truck to travel any distance, you would have covered that same distance and encounter the next truck.

When counting while driving you overestimate the number of trucks and you should compensate by multiplying the count by the average speed of the trucks divided by the combined speed of a truck and your car.

Equation

So with this knowledge it is straight forward to derive an equation:

$$ T_h = T_5 * 12 * V_{rel} $$

The number of trucks per hour equals twelve times the number of trucks counted in a 5 minute period times the relative speed.

$$ {V_{truck} \above{2pt} V_{truck}+V_{car}} $$

This relative speed is the average speed of the trucks divided by the combined speed of a truck and your car.

I have made a short explainer video to illustrate the concepts visually:

Conclusion

So can we estimate the number of trucks per hour passing a fixed point while driving car?

Yes we can, just count them and multiply by the ratio of the average truck speed and the combined speed of a truck and our car.

Be aware that this is just an estimate, as trucks might drive slightly faster and not exactly all at the same speed. They are likely also not driving all at the same distance from each other, but for larger numbers or are larger counting period this all evens out. Just keep your own speed as constant as possible, and keep an eye on the road 🫣

Links

Apparently traffic experts back in the day were just as lazy (or 'efficient') as me and considered counting vehicles from a moving car in considerable detail, as can be seen in this old publication.

And if you want to know more about counting traffic in general, Wikipedia is always a good start.

Acknowledgements

Image at the top was created by Gemini Nano Banana. I only had to tell it twice that trucks are not allowed on the third lane.