## Tuesday, April 7, 2015

### [OlymPri20150402RLD] Peter, James and the Dog

Question

Introduction
This is a cute primary school olympiad type of question.  I vaguely remember reading a similar quiz question from Readers’ Digest (?) many years ago about some bumble bee flying between two trains going towards each other.  Or something like that.
If you try to solve this using “advanced mathematics” like using the sum of two infinite geometric series you can get the answer, but it is quite a swirling mess, like this:-

Another way to slice the dice
Is there a short cut?  Yes!  That requires thinking about the problem in another way.  Notice that the dog’s speed is the sum of Peter’s and James’ speeds.  Imagine ... if James is not moving, but Peter is running at  3 ms-1,  from Peter’s point of view the Earth would be pushed backwards at 3 ms-1  and James would appear to be going towards him at  3 ms-1.  But James is running towards Peter at  2 ms-1, so from Peter’s point of view, it seems that James is coming towards him at  5 ms-1.  Likewise, from James’ point of view, Peter appears to be coming at him at  5 ms-1.  This is the concept of relative speed.  Notice also that the relative distance (gap) between James and Peter is reducing at this speed.  This is because at Peters’ end the gap is reduced at 3 ms-1  and at James’ end the distance is reduced by  2 ms-1, giving a total gap-reduction speed of  5 ms-1.  With these perceptive observations, the answer falls straight out.

Solution
Since the dog’s speed (5 ms-1) is the sum of Peter’s speed (3 ms-1) and James’ speed (2 ms-1), it is always covering a distance at a speed which is the same as the speed of the closing of the gap between Peter and James.  Hence the total distance travelled by the dog must be the same as the initial gap, which is  1 km.

Moral of the Story
Sometimes, you do not need advanced maths, but acute observations.
 When two entities are moving towards each other, their relative speed is the sum of their speeds.  This is also the same as the rate at which the relative distance (gap between the two) is closing.

Suitable Levels