Brain-Teaser 752: Traffic cop
From The Sunday Times, 14th December 1975 [link]
Some years ago I was lecturing to a group of policeman and I set them a traffic problem which they found difficult. Here it is:
A column of vehicles 10 miles long drives 24 miles at a constant speed and then halts. A policeman on motor-cycle starts at the back of the column as it moves off, he rides to the front, turns round immediately and rides to the back of the column arriving at the moment the vehicles halt.
Assuming that his speed has been constant throughout: how far has he ridden?
This puzzle is included in the book The Sunday Times Book of Brain-Teasers: Book 1 (1980). The puzzle text above is taken from the book.
[teaser752]
John Crabtree and 
S2T2 
Jim Randell 8:23 am on 4 May 2021 Permalink |
The front of the convoy starts at a distance of 24 miles from the finish. And ends at distance zero.
The back of the convoy starts at a distance of 34 miles, and ends at a distance of 10 miles.
If we consider that the motorbike also starts at a distance of 34 miles, goes towards the finish until a distance x miles away (0 < x < 24), and then turns around and continues until it reaches the finish point of the end of the convoy. The distance travelled by the motorbike is:
And in the time taken for the motor bike to travel to the turnaround point the front of the convoy has travelled a distance: (24 − x) miles.
And after the motorbike has turned around the convoy travels the remaining distance x miles.
The speeds are constant, so the ratio of the convoy : motorbike distances must be the same in the two parts:
So the turnaround point was 4 miles before the finish, and:
Solution: The motorbike travelled a total distance of 36 miles.
The two parts of the journey are:
Before turnaround: Convoy travels from 24 miles away to 4 miles away; distance = 20 miles. Motorbike travels from 34 miles away to 4 miles away; distance = 30 miles.
After turnaround: Convoy travels from 4 miles away to finish; distance = 4 miles. Motorbike travels from 4 miles away to 10 miles away; distance = 6 miles.
And we see that the motorbike is travelling at 1.5× the speed of the convoy.
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John Crabtree 10:40 pm on 4 May 2021 Permalink |
Let m be the motor-cycle speed and c be the convoy speed.
The distance travelled by the policeman = d = 24m/c
Considering the travel times, 10/(m – c) + 10/(m + c) = 24/c
20mc = 24(m^2 – c^2)
d^2 – 20d – 24^2 = 0
d = 10 +/- 26 = 36 miles.
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