## Driving to Denver

This is really a very simple problem, but many people find it very counter-intuitive.

You are driving by car to a particular destination, and our only assumption is that you are free to drive at any speed you choose - no traffic jams or anything like that. For the first half of the journey (and by half we mean half the overall distance between the starting point and your finishing point) you drive at 20 miles per hour. You then realise that this is all taking much too long, and that you are going to be late. You therefore decide that you will increase your speed so that your overall average speed for the whole journey will be 40 miles per hour. How fast do you have to drive for the remaining part of your journey in order for your average speed for the whole journey to be 40 miles per hour? Strangely enough the answer is the same, whatever the total distance of your journey.

The answer is on the next page.

One of the things that people do sometimes find difficult is that you have not specified the distance, but in fact the answer is the same, whatever the distance. The answer is that there is no answer - there is no speed you can drive at that will make your average speed 40 miles per hour. The reason is that to make your average speed for the whole journey 40 miles per hour, you need to be at your destination NOW, and no speed will allow you to achieve this (other than crossing to a place where the space-time continuum breaks down!)

How does this come about? Well, suppose you started out at 2 o'clock, and your destination is exactly 40 miles away. For your average speed to be 40 miles per hour, you need to arrive at 3 o'clock. But you drove the first half of the journey, 20 miles in this case, at 20 miles per hour. Which has taken you an hour. So it is now 3 o'clock, and you have only covered half the distance. There is no speed that you can drive for the second half of the journey that will make the average speed 40 miles per hour, since you need to be at your destination at this moment. And this applies no matter what the distance of the journey. If you prefer, you can change everything to kilometres per hour, instead of miles per hour.