Jocelyn Pollard, of Dunedin, asks :-
As a motorist traveling regularly up and down between Dunedin and Christchurch, I have often pondered the question of how long a rest stop one would need to avoid ever catching up again with the oversize traffic ahead. For example, I recently became stuck behind a line of sheep truck, milk tanker, private car and a truck and trailer unit. The convoy speed was 80 to 100kph. The vehicles were too close together to pass individually. Fortunately, they all passed the Waimate turn-off cafe, so I stopped for a coffee. Sure enough, I didn't catch up with them again. Assuming that all were traveling to the same destination, is there an underlying rule of thumb about the time I need to stop for to ensure I don't get held up again?
Astrid an Huef, a mathematician at the University of Otago, responded.
Let's assume that there is not too much variation in the speed of the convoy, and that its average speed is s km/hour. Since we don't encourage speeding, let's assume that you want to travel only 5km/hour faster than the convoy.
Average speed is distance travelled divided by the time taken, and the time taken to travel a distance d at an average speed s is d divided by s. If you stop a distance d km from Christchurch, then the minimum time you should wait is the time it would take you to catch up with the convoy just when it arrives in Christchurch. So the time to wait should be roughly the difference between the time for the convoy at average speed s to reach Christchurch and the time for you at average speed (s + 5) to reach Christchurch.
So the time you should wait is roughly d/s minus d/(s + 5) = ((s + 5)d - sd) divided by s(s + 5) = 5d divided by (s(s + 5)).
Specifically, Waimate is about 200 km from Christchurch. Assuming the convoy has an average speed of 90 km/hr and you want to travel with an average speed of 95 km/hr then you should wait at least 5 times 200 divided by (90 times 95) hours, that is, approximately 0.117 hours or 7 minutes.
As heavy trucks usually slow down considerably when going up hills, if you are enjoying your coffee you should give it another 5 minutes to be safe (but then again, the next convoy could just be crawling past as you get back in the car).