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+----- Thread: The Voorhees law of traffic (/showthread.php?tid=2189)
The Voorhees law of traffic - mklabgr - 09-25-2026
The Guardian article examines a familiar driving experience: you overtake a slower car, only to encounter a red traffic light and find that the slower vehicle catches up with you. Mathematics researcher Dr Conor Boland of Dublin City University studied this phenomenon and called it the “Voorhees law of traffic”, a humorous reference to Jason Voorhees, the slow-moving horror character who nevertheless seems to catch his victims. The research was published in Royal Society Open Science.
Boland's model considers vehicles travelling along a road containing traffic lights operating on fixed cycles. Even when one driver travels faster than another, the faster driver does not necessarily arrive at the destination earlier because the traffic-light timing introduces delays. If the first driver reaches a red light while the slower driver arrives during a green phase, their apparent advantage can disappear. Mathematically, the difference in arrival times depends not only on speed but also on the phase of the traffic lights relative to each vehicle's position.
The interesting result is that, for a single traffic light, the apparent tendency of the slower car to “catch up” is largely a consequence of probability and perception rather than a deterministic law. Over many journeys, gains from travelling faster and losses caused by red lights can approximately balance. If there are several independently timed traffic lights, however, the probability of the faster driver being stopped while the slower driver proceeds increases. In simplified terms, if the probability of being delayed at one light is $p$, then the probability of avoiding that delay at $n$ independent lights is $(1-p)^n$.
The research therefore provides a neat example of how mathematics can explain an everyday phenomenon that feels almost impossible. It also suggests that speeding or overtaking does not guarantee a meaningful reduction in journey time when traffic signals dominate the journey. The article notes that real-world driving is considerably more complicated than the mathematical model because drivers change speed, lights may be sensor-controlled, and traffic conditions vary. Nevertheless, the study illustrates how probability, geometry and simple dynamical modelling can turn an apparently mysterious experience into something mathematically understandable.
Key takeaways
- The “slow car catches you” phenomenon is mathematically explainable, rather than simply bad luck.
- Traffic lights introduce a discrete timing constraint that can eliminate the advantage of higher speed.
- If two cars travel distances $d$ at speeds $v_1$ and $v_2$, their uninterrupted travel times are
$t_1=\frac{d}{v_1}$ and $t_2=\frac{d}{v_2}$.
- With a traffic-light delay $D$, actual travel time becomes approximately
$T=\frac{d}{v}+D$.
- With multiple independent lights, probability becomes increasingly important; for example, the probability of experiencing at least one delay can be written as
$P(\text{delay})=1-(1-p)^n$.
- The phenomenon demonstrates how human intuition can be misleading when randomness and repeated events are involved.
- Teaching connection: it is an excellent real-world introduction to probability, conditional probability, functions, modelling, expected values and simulation.
- A classroom simulation could randomly assign traffic-light phases to two cars and investigate whether increasing speed actually reduces the expected journey time.
ARTICLE