Point of view switcheroo

Rolling marbles 

Here's a puzzle: Suppose you have N identical marbles rolling along a one dimensional table-top.  Each marble is randomly rolling to the left or to the right, all with the same speed.  Collisions are elastic, which means the marbles just change direction.  What is the maximum amount of time before all the marbles have rolled off the table?

8 marbles with speed 1 on a table of length 1

Answering this question is really difficult if you simply pick an individual marble and try to work out how long it might stay on the table as it bounces back and forth.  But there's a simpler way to look at it.

Prior to each collision you have one marble rolling to the left and one to the right, and afterwards you still have one rolling to the left and one to the right.  If we swap labels following each collision then the labels never change direction.  Now it's easy to see that the answer is the same whether there are 100 marbles or just one.  And all we did was switch our PoV from the masses to the momenta.

 

Replacing your petrol car with a clean green EV

We can apply a similar point of view switcheroo to other questions too.  For example, should you sell your petrol car and buy an electric one?  The trick here is to switch from thinking about your emissions to that of the car.  There are an approximately constant number of cars in the UK, meaning that when one reaches it's end of life (at perhaps 150,000 miles) it is scrapped and replaced by a new car on the UK's roads.  Viewed this way, the only thing that makes any difference is whether or not the new cars being inserted into the pipeline are electric.  So if you are buying a brand new car you can help by making sure it is electric, but if you normally buy cars second hand there's not much difference you can make without giving up driving altogether.

So, whilst ordinary people can't make much of a difference in this area, the government could easily make a very big difference.  For example, by introducing a rule that all new sales must be electric after a given date, like Boris Johnson's government did.  The worst thing it could do is to set that date back, like Rishi Sunak's government has.  And if they were really serious they could bring the date forward and introduce an early scrappage scheme for polluting cars.  Because even the most environmentally conscious driver is not going to scrap a petrol car they could otherwise sell.