Teaser 2508: [Near miss]
From The Sunday Times, 17th October 2010 [link] [link]
An Austin was pootling along a country lane at 30mph; behind were a Bentley doing 40mph and a Cortina doing 50mph. The Bentley and the Cortina braked simultaneously, decelerating at constant rates, while the Austin carried on at the same speed. The Bentley just avoided hitting the rear of the Austin, [while, at the same time,] the Cortina just avoided a collision with the Bentley. The Bentley and the Cortina continued to decelerate at the same rates, and stopped with a 45 yard gap between them.
What was the gap between the Bentley and the Cortina at the moment they started to brake?
The wording in this puzzle has been modified from the published version for clarification.
This puzzle was originally published with no title.
[teaser2508]
S2T2 
Jim Randell 8:46 am on 29 December 2020 Permalink |
I am assuming that at the time of the “almost collision” the separation between A, B and B, C can be considered to be zero. (And so we can consider the cars to be points, that coincide at the moment of “almost collision”).
This puzzle can be solved graphically, without the need to resort to equations of motion (although you can solve it that way too).
If we plot the velocities of the Austin (red), the Bentley (green), and the Cortina (blue) against time we get a graph like this:
Where red carries on at a steady 30mph, green starts at 40mph and decelerates steadily to 0mph (BB’), and blue starts at 50mph and decelerates steadily to 0mph (CC’).
At X, all their speeds are 30mph, and this is the point at which the separations between the cars are zero (the almost collision).
The area under the line XB’ gives the distance travelled by green after the almost collision, and the area under the line XC’ gives the distance travelled by blue after the almost collision.
And the difference between these distances corresponds to their final separation:
Similarly we can calculate the difference between the areas under the lines CX and BX to get the separation of green and blue at the time they started braking:
Solution: The Bentley and the Cortina were 10 yards apart at the time they started braking.
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